Search: a059344 -id:a059344
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A118393
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Eigenvector of triangle A059344. E.g.f.: exp( Sum_{n>=0} x^(2^n) ).
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+20
3
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1, 1, 3, 7, 49, 201, 1411, 7183, 108417, 816049, 9966691, 80843511, 1381416433, 14049020857, 216003063459, 2309595457471, 72927332784001, 1046829280528353, 23403341433961027, 329565129021010279, 9695176730057249841, 160632514329660035881
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OFFSET
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0,3
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COMMENTS
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E.g.f. of A059344 is: exp(x+y*x^2). More generally, given a triangle with e.g.f.: exp(x+y*x^b), the eigenvector will have e.g.f.: exp( Sum_{n>=0} x^(b^n) ).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..[n/2]} n!/k!/(n-2*k)! *a(k) for n>=0, with a(0)=1.
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MAPLE
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option remember;
if n <=1 then
1;
else
n!*add(procname(k)/k!/(n-2*k)!, k=0..n/2) ;
end if;
end proc:
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((j-> j!*
a(n-j)*binomial(n-1, j-1))(2^i), i=0..ilog2(n)))
end:
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MATHEMATICA
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a[0] = 1; a[n_] := a[n] = Sum[n!/k!/(n - 2*k)!*a[k], {k, 0, n/2}];
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PROG
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(PARI) a(n)=n!*polcoeff(exp(sum(k=0, #binary(n), x^(2^k))+x*O(x^n)), n)
(Sage)
f=factorial;
def a(n): return 1 if n==0 else sum((f(n)/(f(k)*f(n-2*k)))*a(k) for k in (0..n//2))
(Magma)
function a(n)
if n eq 0 then return 1;
else return (&+[ (Factorial(n)/(Factorial(k)*Factorial(n-2*k)))*a(k): k in [0..Floor(n/2)]]);
end if; return a; end function;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A001813
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Quadruple factorial numbers: a(n) = (2n)!/n!.
(Formerly M2040 N0808)
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+10
140
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1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000, 830034394580628357120000, 58102407620643984998400000
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OFFSET
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0,2
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COMMENTS
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Counts binary rooted trees (with out-degree <= 2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.
Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003
Number of broadcast schemes in the complete graph on n+1 vertices, K_{n+1}. - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - Wolfdieter Lang, Jan 09 2012
Aerated with intervening zeros (1,0,2,0,12,0,120,...) = a(n) (cf. A123023 and A001147), the e.g.f. is e^(t^2), so this is the base for the Appell sequence with e.g.f. e^(t^2) e^(x*t) = exp(P(.,x),t) (reverse A059344, cf. A099174, A066325 also). P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for e^(-t^2)e^(x*t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), e.g., (P(.,t))^n = P(n,t).
Equals A000407*2 with leading 1 added. (End)
a(n) is also the number of square roots of any permutation in S_{4*n} whose disjoint cycle decomposition consists of 2*n transpositions. - Luis Manuel Rivera Martínez, Mar 04 2015
For n > 1, it follows from the formula dated Aug 07 2013 that a(n) is a Zumkeller number (A083207). - Ivan N. Ianakiev, Feb 28 2017
For n divisible by 4, a(n/4) is the number of ways to place n points on an n X n grid with pairwise distinct abscissae, pairwise distinct ordinates, and 90-degree rotational symmetry. For n == 1 (mod 4), the number of ways is a((n-1)/4) because the center point can be considered "fixed". For 180-degree rotational symmetry see A006882, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, Eq. 32.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
Eugene McDonnell, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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E.g.f.: (1-4*x)^(-1/2).
a(n) = (2*n)!/n! = Product_{k=0..n-1} (4*k + 2).
Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n) = int(x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)), x=0..infinity), n=0, 1, .. . This representation is unique. - Karol A. Penson, Sep 18 2001
Define a'(1)=1, a'(n) = Sum_{k=1..n-1} a'(n-k)*a'(k)*C(n, k); then a(n)=a'(n+1). - Benoit Cloitre, Apr 27 2003
With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry, May 09 2003
a(n) = A000680(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (4*i + 2) = 4^n*Pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
For asymptotics, see the Robinson paper.
a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);
G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). - Paul Barry, Nov 25 2009
a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
2, 2, 0, 0, 0, 0, ...
4, 4, 4, 0, 0, 0, ...
6, 6, 6, 6, 0, 0, ...
8, 8, 8, 8, 8, 0, ...
...
(End)
a(n) = (-2)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0), where Q(k) = 1 + x*(4*k+2) - x*(4*k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 8*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
D-finite with recurrence: a(n) = (4*n-6)*a(n-2) + (4*n-3)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 07 2013
Sum_{n>=0} 1/a(n) = (exp(1/4)*sqrt(Pi)*erf(1/2) + 2)/2 = 1 + A214869, where erf(x) is the error function. - Ilya Gutkovskiy, Nov 10 2016
Sum_{n>=0} (-1)^n/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)), where erfi(x) is the imaginary error function. - Amiram Eldar, Feb 20 2021
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EXAMPLE
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The following permutations of order 8 and their reversals have this property:
1 7 3 5 2 4 0 6
1 7 4 2 5 3 0 6
2 3 7 6 1 0 4 5
2 4 7 1 6 0 3 5
3 2 6 7 0 1 5 4
3 5 1 7 0 6 2 4
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MAPLE
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A001813 := n -> mul(k, k = select(k-> k mod 4 = 2, [$1 .. 4*n])): seq(A001813(n), n=0..16);
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MATHEMATICA
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PROG
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(Sage) [binomial(2*n, n)*factorial(n) for n in range(0, 17)] # Zerinvary Lajos, Dec 03 2009
(PARI) first(n) = x='x+O('x^n); Vec(serlaplace((1 - 4*x)^(-1/2))) \\ Iain Fox, Jan 01 2018 (corrected by Iain Fox, Jan 11 2018)
(Maxima) makelist(binomial(n+n, n)*n!, n, 0, 30); /* Martin Ettl, Nov 05 2012 */
(Magma) [Factorial(2*n)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 09 2018
(GAP) List([0..20], n->Factorial(2*n)/Factorial(n)); # Muniru A Asiru, Nov 01 2018
(Python)
from math import factorial
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CROSSREFS
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Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence), A000165, A047055, A047657, A084947, A084948, A084949, A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.
Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A059343
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Triangle of nonzero coefficients of Hermite polynomials H_n(x) in increasing powers of x.
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+10
12
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1, 2, -2, 4, -12, 8, 12, -48, 16, 120, -160, 32, -120, 720, -480, 64, -1680, 3360, -1344, 128, 1680, -13440, 13440, -3584, 256, 30240, -80640, 48384, -9216, 512, -30240, 302400, -403200, 161280, -23040, 1024, -665280, 2217600, -1774080, 506880, -56320, 2048, 665280, -7983360, 13305600
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OFFSET
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0,2
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 50.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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EXAMPLE
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1; 2*x; -2+4*x^2; -12*x+8*x^3; ...
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MAPLE
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with(orthopoly): h:=proc(n) if n mod 2=0 then expand(x^2*H(n, x)) else expand(x*H(n, x)) fi end: seq(seq(coeff(h(n), x^(2*k)), k=1..1+floor(n/2)), n=0..14); # this gives the signed sequence
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MATHEMATICA
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Flatten[ Table[ Coefficient[ HermiteH[n, x], x, k], {n, 0, 12}, {k, Mod[n, 2], n, 2}]] (* Jean-François Alcover, Jan 23 2012 *)
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PROG
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(Python)
from sympy import hermite, Poly, Symbol
x = Symbol('x')
def a(n):
return Poly(hermite(n, x), x).coeffs()[::-1]
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CROSSREFS
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If initial zeros are included, same as A060821.
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KEYWORD
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sign,easy,nice,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A119275
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Inverse of triangle related to Padé approximation of exp(x).
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+10
5
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1, -2, 1, 0, -6, 1, 0, 12, -12, 1, 0, 0, 60, -20, 1, 0, 0, -120, 180, -30, 1, 0, 0, 0, -840, 420, -42, 1, 0, 0, 0, 1680, -3360, 840, -56, 1, 0, 0, 0, 0, 15120, -10080, 1512, -72, 1, 0, 0, 0, 0, -30240, 75600, -25200, 2520, -90, 1, 0, 0, 0, 0, 0, -332640, 277200, -55440, 3960, -110, 1
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OFFSET
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0,2
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COMMENTS
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Row sums are (-1)^(n+1)*A000321(n+1).
Also the inverse Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) giving unsigned values and adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
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LINKS
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FORMULA
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T(n,k) = [k<=n]*(-1)^(n-k)*(n-k)!*C(n+1,k+1)*C(k+1,n-k).
E.g.f.: exp(x*(t-t^2)) - 1 = x*t + (-2*x+x^2)*t^2/2! + (-6*x^2+x^3)*t^3/3! + (12*x^2-12*x^3+x^4)*t^4/4! + .... Cf. A059344. Let D denote the operator sum {k >= 0} (-1)^k/k!*x^k*(d/dx)^(2*k). The n-th row polynomial R(n,x) = D(x^n) and satisfies the recurrence equation R(n+1,x) = x*R(n,x)-2*n*x*R(n-1,x). The e.g.f. equals D(exp(x*t)).
(End)
With initial index n = 1 and unsigned, these are the partition row polynomials of A130561 and A231846 with c_1 = c_2 = x and c_n = 0 otherwise. The first nonzero, unsigned element of each diagonal is given by A001813 (for each row, A001815) and dividing along the corresponding diagonal by this element generates A098158 with its first column removed (cf. A034839 and A086645).
The n-th polynomial is generated by (x - 2y d/dx)^n acting on 1 and then evaluated at y = x, e.g., (x - 2y d/dx)^2 1 = (x - 2y d/dx) x = x^2 - 2y evaluated at y = x gives p_2(x) = -2x + x^2.
(End)
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EXAMPLE
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Triangle begins
1,
-2, 1,
0, -6, 1,
0, 12, -12, 1,
0, 0, 60, -20, 1,
0, 0, -120, 180, -30, 1,
0, 0, 0, -840, 420, -42, 1,
0, 0, 0, 1680, -3360, 840, -56, 1,
0, 0, 0, 0, 15120, -10080, 1512, -72, 1
Row 4: D(x^4) = (1 - x*(d/dx)^2 + x^2/2!*(d/dx)^4 - ...)(x^4) = x^4 - 12*x^3 + 12*x^2.
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MAPLE
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# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n<2, (n+1)*(-1)^n, 0), 9); # Peter Luschny, Jan 27 2016
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MATHEMATICA
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Table[(-1)^(n - k) (n - k)!*Binomial[n + 1, k + 1] Binomial[k + 1, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 12 2016 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[#<2, (#+1) (-1)^#, 0]&, rows];
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PROG
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(Sage) # uses[inverse_bell_matrix from A265605]
# Unsigned values and an additional first column (1, 0, 0, ...).
multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
inverse_bell_matrix(multifact_4_2, 9) # Peter Luschny, Dec 31 2015
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CROSSREFS
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Cf. A059344 (unsigned row reverse).
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KEYWORD
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AUTHOR
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STATUS
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approved
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A122832
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Exponential Riordan array (e^(x(1+x)),x).
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+10
4
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1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 25, 28, 18, 4, 1, 81, 125, 70, 30, 5, 1, 331, 486, 375, 140, 45, 6, 1, 1303, 2317, 1701, 875, 245, 63, 7, 1, 5937, 10424, 9268, 4536, 1750, 392, 84, 8, 1, 26785, 53433, 46908, 27804, 10206, 3150, 588, 108, 9, 1
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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Number triangle T(n,k) = (n!/k!)*Sum_{i = 0..n-k} C(i,n-k-i)/i!.
Array is exp(S + S^2) where S is A132440 the infinitesimal generator for Pascal's triangle.
T(n,k) = binomial(n,k)*A047974(n-k).
So T(n,k) gives the number of ways to choose a subset of {1,2,...,n) of size k and then arrange the remaining n-k elements into a set of lists of length 1 or 2. (End)
n-th row polynomial: R(n,x) = exp(D + D^2) (x^n) = exp(D^2) (1 + x)^n, where D denotes the derivative operator d/dx. Cf. A111062.
The sequence of polynomials defined by R(n,x-1) = exp(D^2) (x^n) begins [1, 1, 2 + x^2, 6*x + x^3, 12 + 12*x^2 + x^4, ...] and is related to the Hermite polynomials. See A059344. (End)
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EXAMPLE
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Triangle begins:
1;
1, 1;
3, 2, 1;
7, 9, 3, 1;
25, 28, 18, 4, 1;
81, 125, 70, 30, 5, 1;
...
T(3,1) = 9. The 9 ways to select a subset of {1,2,3} of size 1 and arrange the remaining elements into a set of lists (denoted by square brackets) of length 1 or 2 are:
{1}[2,3], {1}[3,2], {1}[2][3],
{2}[1,3], {2}[3,1], {2}[1][3],
{3}[1,2], {3}[2,1], {3}[1][2]. (End)
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MATHEMATICA
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(* The function RiordanArray is defined in A256893. *)
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PROG
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(PARI) T(n, k) = (n!/k!)*sum(i=0, n-k, binomial(i, n-k-i)/i!); \\ Michel Marcus, Aug 28 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A118394
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Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.
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+10
3
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1, 1, 1, 1, 6, 1, 24, 1, 60, 1, 120, 360, 1, 210, 2520, 1, 336, 10080, 1, 504, 30240, 60480, 1, 720, 75600, 604800, 1, 990, 166320, 3326400, 1, 1320, 332640, 13305600, 19958400, 1, 1716, 617760, 43243200, 259459200, 1, 2184, 1081080, 121080960, 1816214400
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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E.g.f.: A(x,y) = exp(x + y*x^3).
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EXAMPLE
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Triangle begins:
1;
1;
1;
1, 6;
1, 24;
1, 60;
1, 120, 360;
1, 210, 2520;
1, 336, 10080;
1, 504, 30240, 60480;
1, 720, 75600, 604800;
1, 990, 166320, 3326400;
1, 1320, 332640, 13305600, 19958400;
...
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MATHEMATICA
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T[n_, k_] := n!/(k!(n-3k)!);
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PROG
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(PARI) T(n, k)=if(n<3*k || k<0, 0, n!/k!/(n-3*k)!)
(Sage)
f=factorial;
flatten([[f(n)/(f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021
(Magma)
F:= Factorial;
[F(n)/(F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // G. C. Greubel, Mar 07 2021
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A113216
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Triangle of polynomials P(n,x) of degree n related to Pi (see comment) and derived from Padé approximation to exp(x).
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+10
1
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1, 1, 2, 1, -6, -12, 1, 12, -60, -120, 1, -20, -180, 840, 1680, 1, 30, -420, -3360, 15120, 30240, 1, -42, -840, 10080, 75600, -332640, -665280, 1, 56, -1512, -25200, 277200, 1995840, -8648640, -17297280, 1, -72, -2520, 55440, 831600, -8648640, -60540480, 259459200, 518918400, 1, 90, -3960, -110880
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OFFSET
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0,3
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COMMENTS
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P(n,x) is a sequence of polynomials of degree n with integer coefficients, having exactly n real roots, such that r(n) the smallest root (in absolute value) converges quickly to Pi/2. e.g. the appropriate root for P(5,x) is r(5)=1.5707963(4026....) . To see the rapidity of convergence it is relevant noting that (r(n)-Pi/2)(2n)! -->0 as n grows.
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LINKS
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FORMULA
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P(0, x) = 1, P(1, x) = x+2, P(n, x) = (4*n-2)*P(n-1, x)-x^2*P(n-2, x).
P(n, x) = Sum_{0<=i<=n} (-1)^floor(i/2)*(2n-i)!/i!/(n-i)!*x^i.
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EXAMPLE
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P(5,x) = x^5 + 30*x^4 - 420*x^3 - 3360*x^2 + 15120*x + 30240.
Triangle begins:
1;
1,2;
1,-6,-12;
1,12,-60,-120;
1,-20,-180,840,1680;
1,30,-420,-3360,15120,30240;
1,-42,-840,10080,75600,-332640,-665280;
...
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PROG
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(PARI) P(n, x)=if(n<2, if(n%2, x+2, 1), (4*n-2)*P(n-1, x)-x^2*P(n-2, x))
(PARI) P(n, x)=sum(i=0, n, x^i*(-1)^floor(i/2)/(n-i)!/i!*(2*n-i)!)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A135610
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Triangle read by rows: the k-th entry of row n is the number of particular connectivity requirements that a k-linked graph with n >= 2k vertices has to satisfy T(n,k) = (1/2) * n!/(k!*(n-2*k)!) where k runs from 1 to floor(n/2).
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+10
1
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1, 3, 6, 6, 10, 30, 15, 90, 60, 21, 210, 420, 28, 420, 1680, 840, 36, 756, 5040, 7560, 45, 1260, 12600, 37800, 15120, 55, 1980, 27720, 138600, 166320, 66, 2970, 55440, 415800, 997920, 332640, 78, 4290, 102960, 1081080, 4324320, 4324320, 91, 6006
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OFFSET
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1,2
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COMMENTS
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A graph G with n >= 2k vertices is k-linked if and only if for every choice of two disjoint sets of vertices, each having cardinality k, and for every numbering {s_1,...,s_k} and {t_1,...,t_k} of the vertices in the respective sets, there exist k disjoint paths P_1,...,P_k in G such that P_j starts at s_j and ends at t_j. Note that k is at most floor(n/2).
Being k-linked is a considerably stronger property of graph than being merely k-connected and this sequence is a superficial quantitative answer to the question of how much stronger a property it is. For graph-theoretical answers to how connectedness and linkedness are related, see the references.
The sequence arises as follows. Let a particular connectivity requirement consist of a choice of two vertex sets as described above, together with a particular numbering of the respective elements in the sets, i.e., together with a prescription which vertex has to be linked to which. The set of all such particular connectivity requirements can be constructed as follows.
First pick a set of k vertices, then pick another set of vertices among the remaining n-k vertices of G and then, going along the vertices in the first set in an arbitrary order, successively prescribe what vertex it is to be linked to. For the first prescription there are k possibilities, for the next k-1, etc., so clearly, since there are no dependencies, for a fixed choice of two vertex sets there are k! prescriptions. It is clear that in this process every possible particular connectivity requirement arises exactly twice, since there are two possibilities to choose the first of the two k-element sets. Thus there are (1/2) * C(n,k) * C(n-k, k) * k! = (1/2) * n!/(k!*(n-2*k)!) particular connectivity requirements.
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REFERENCES
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R. Diestel, Graph Theory, 3rd edition, Springer 2005 (Chapter 3.5).
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LINKS
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FORMULA
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T(n, k) = (1/2) * n!/(k!*(n-2*k)!).
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EXAMPLE
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If n=4 and k=1, then (1/2)*C(4,1)*C(4-1,1)*1! = 6, so there are 6 particular connectivity requirements that a 1-linked graph with 4 vertices has to satisfy.
If n=4 and k=2, then (1/2)*C(4,2)*C(4-2,2)*2! = 6, so there are again 6 particular connectivity requirements that a 2-linked graph with 4 vertices has to satisfy.
Triangle begins:
1;
3;
6, 6;
10, 30;
15, 90, 60;
21, 210, 420;
28, 420, 1680, 840;
36, 756, 5040, 7560;
45, 1260, 12600, 37800, 15120;
..
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MAPLE
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seq(seq(n!/(k!*(n-2*k)!)/2, k=1..floor(n/2)), n=1..20);
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CROSSREFS
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This is A059344/2 without column k=0.
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KEYWORD
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nonn,tabf
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AUTHOR
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Peter C. Heinig (algorithms(AT)gmx.de), Feb 27 2008
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STATUS
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approved
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