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Search: a059344 -id:a059344
Displaying 1-8 of 8 results found. page 1
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A118393 Eigenvector of triangle A059344. E.g.f.: exp( Sum_{n>=0} x^(2^n) ). +20
3
1, 1, 3, 7, 49, 201, 1411, 7183, 108417, 816049, 9966691, 80843511, 1381416433, 14049020857, 216003063459, 2309595457471, 72927332784001, 1046829280528353, 23403341433961027, 329565129021010279, 9695176730057249841, 160632514329660035881 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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) ).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..450
FORMULA
a(n) = Sum_{k=0..[n/2]} n!/k!/(n-2*k)! *a(k) for n>=0, with a(0)=1.
MAPLE
A118393 := proc(n)
option remember;
if n <=1 then
1;
else
n!*add(procname(k)/k!/(n-2*k)!, k=0..n/2) ;
end if;
end proc:
seq(A118393(n), n=0..20) ; # R. J. Mathar, Aug 19 2014
# 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:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 01 2017
MATHEMATICA
a[0] = 1; a[n_] := a[n] = Sum[n!/k!/(n - 2*k)!*a[k], {k, 0, n/2}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 18 2018 *)
PROG
(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))
[a(n) for n in (0..25)] # G. C. Greubel, Feb 18 2021
(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;
[a(n): n in [0..25]]; // G. C. Greubel, Feb 18 2021
CROSSREFS
Cf. A059344, variants: A118395, A118930.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 07 2006
STATUS
approved
A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.
(Formerly M2040 N0808) 		+10
140
1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000, 830034394580628357120000, 58102407620643984998400000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
Hankel transform is A137565. - Paul Barry, Nov 25 2009
The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - Wolfdieter Lang, Jan 09 2012
From Tom Copeland, Nov 15 2014: (Start)
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
Self-convolution gives A076729. - Vladimir Reshetnikov, Oct 11 2016
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
REFERENCES
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).
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..100
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Rosa Maria Pidatella, Hermite and Laguerre Functions: a Unifying Point of View, Università degli Studi di Catania, Sicily, Italy (2019).
Murray Bremner and Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - N. J. A. Sloane, Apr 18 2014
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Elliot J. Carr and Matthew J. Simpson, New homogenization approaches for stochastic transport through heterogeneous media, arXiv:1810.08890 [physics.bio-ph], 2018.
W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, Parity reversing involutions on plane trees and 2-Motzkin paths, arXiv:math/0503300 [math.CO], 2005.
Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 8.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 127
S. Goodenough and C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.
S. Goodenough and C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16,
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 115
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets, arXiv:math/0612572 [math.CO], 2006.
Calin D. Morosan, On the number of broadcast schemes in networks, Information Processing Letters, Volume 100, Issue 5 (2006), 188-193.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
Index to divisibility sequences
Index entries for related partition-counting sequences
FORMULA
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(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1). - André F. Labossière, Jun 21 2007
a(n) = 12*A051618(a) n >= 2. - Zerinvary Lajos, Feb 15 2008
a(n) = A000984(n)*A000142(n). - Zerinvary Lajos, Mar 25 2008
a(n) = A016825(n-1)*a(n-1). - Roger L. Bagula, Sep 17 2008
a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
From Paul Barry, Jan 15 2009: (Start)
G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);
a(n) = (n+1)!*A000108(n). (End)
a(n) = Sum_{k=0..n} A132393(n,k)*2^(2n-k). - Philippe Deléham, Feb 10 2009
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) = A173333(2*n,n) for n>0; cf. A006963, A001761. - Reinhard Zumkeller, Feb 19 2010
From Gary W. Adamson, Jul 19 2011: (Start)
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
EXAMPLE
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
MAPLE
A001813 := n->(2*n)!/n!;
A001813 := n -> mul(k, k = select(k-> k mod 4 = 2, [$1 .. 4*n])): seq(A001813(n), n=0..16);
# Peter Luschny, Jun 23 2011
MATHEMATICA
Table[(2n)!/n!, {n, 0, 20}] (* Harvey P. Dale, May 02 2011 *)
PROG
(Sage) [binomial(2*n, n)*factorial(n) for n in range(0, 17)] # Zerinvary Lajos, Dec 03 2009
(PARI) a(n)=binomial(n+n, n)*n! \\ Charles R Greathouse IV, Jun 15 2011
(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
def A001813(n): return factorial(n<<1)//factorial(n) # Chai Wah Wu, Feb 14 2023
CROSSREFS
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.
Cf. A123023, A059344, A099174, A066325, A001700, A000407, A006882.
Cf. A000085, A135401, A297708. - Manfred Scheucher, Jan 07 2018
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from James A. Sellers, May 01 2000
STATUS
approved
A059343 Triangle of nonzero coefficients of Hermite polynomials H_n(x) in increasing powers of x. +10
12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
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.
LINKS
T. D. Noe, Rows n=0..100 of triangle, flattened
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].
P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Eric Weisstein's World of Mathematics, Hermite Polynomial
EXAMPLE
1; 2*x; -2+4*x^2; -12*x+8*x^3; ...
MAPLE
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
MATHEMATICA
Flatten[ Table[ Coefficient[ HermiteH[n, x], x, k], {n, 0, 12}, {k, Mod[n, 2], n, 2}]] (* Jean-François Alcover, Jan 23 2012 *)
PROG
(Python)
from sympy import hermite, Poly, Symbol
x = Symbol('x')
def a(n):
return Poly(hermite(n, x), x).coeffs()[::-1]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017
CROSSREFS
Cf. A059344.
If initial zeros are included, same as A060821.
KEYWORD
sign,easy,nice,tabf
AUTHOR
N. J. A. Sloane, Jan 27 2001
EXTENSIONS
Edited by Emeric Deutsch, Jun 05 2004
STATUS
approved
A119275 Inverse of triangle related to Padé approximation of exp(x). +10
5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Inverse of A119274.
Row sums are (-1)^(n+1)*A000321(n+1).
Bell polynomials of the second kind B(n,k)(1,-2). - Vladimir Kruchinin, Mar 25 2011
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
LINKS
Table of n, a(n) for n=0..65.
Eric Weisstein's World of Mathematics, Bell Polynomial
FORMULA
T(n,k) = [k<=n]*(-1)^(n-k)*(n-k)!*C(n+1,k+1)*C(k+1,n-k).
From Peter Bala, May 07 2012: (Start)
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)
From Tom Copeland, Oct 11 2016: (Start)
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)
EXAMPLE
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.
MAPLE
# 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
MATHEMATICA
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];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)
PROG
(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
CROSSREFS
Cf. A059344 (unsigned row reverse).
Cf. A034839, A001813, A001815, A086645, A098158, A130561, A231846.
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, May 12 2006
STATUS
approved
A122832 Exponential Riordan array (e^(x(1+x)),x). +10
4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Row sums are A000898. Inverse is A122833. Product of A007318 and A067147.
LINKS
Michel Marcus, Rows n=0..50 of triangle, flattened
FORMULA
Number triangle T(n,k) = (n!/k!)*Sum_{i = 0..n-k} C(i,n-k-i)/i!.
From Peter Bala, May 14 2012: (Start)
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)
From Peter Bala, Oct 24 2023: (Start)
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)
EXAMPLE
Triangle begins:
1;
1, 1;
3, 2, 1;
7, 9, 3, 1;
25, 28, 18, 4, 1;
81, 125, 70, 30, 5, 1;
...
From Peter Bala, May 14 2012: (Start)
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)
MATHEMATICA
(* The function RiordanArray is defined in A256893. *)
RiordanArray[E^(#(1+#))&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)
PROG
(PARI) T(n, k) = (n!/k!)*sum(i=0, n-k, binomial(i, n-k-i)/i!); \\ Michel Marcus, Aug 28 2017
CROSSREFS
A000898 (row sums), A047974 (column 0), A291632 (column 1), A122833 (inverse array).
Cf. A059344, A111062.
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Sep 12 2006
EXTENSIONS
More terms from Michel Marcus, Aug 28 2017
STATUS
approved
A118394 Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Row sums form A118395.
Eigenvector is A118396.
LINKS
G. C. Greubel, Rows n = 0..150 of the triangle, flattened
FORMULA
E.g.f.: A(x,y) = exp(x + y*x^3).
EXAMPLE
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;
...
MATHEMATICA
T[n_, k_] := n!/(k!(n-3k)!);
Table[T[n, k], {n, 0, 14}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Nov 04 2020 *)
PROG
(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
CROSSREFS
Cf. A118395 (row sums), A118396 (eigenvector).
Variants: A059344, A118931.
KEYWORD
nonn,tabf
AUTHOR
Paul D. Hanna, May 07 2006
STATUS
approved
A113216 Triangle of polynomials P(n,x) of degree n related to Pi (see comment) and derived from Padé approximation to exp(x). +10
1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..48.
FORMULA
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.
EXAMPLE
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;
...
PROG
(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)!)
CROSSREFS
Cf. A113025 (unsigned variant), A048854, A059344, A119274.
KEYWORD
sign,tabl
AUTHOR
Benoit Cloitre, Jan 07 2006
STATUS
approved
A135610 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). +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
REFERENCES
R. Diestel, Graph Theory, 3rd edition, Springer 2005 (Chapter 3.5).
LINKS
Peter C. Heinig, Table of n, a(n) for n = 1..100
D. Kuhn and D. Osthus, Topological minors in graphs of large girth, J. Comb. Theory B 86 (2002), 364-380.
W. Mader, Topological subgraphs in graphs of large girth, Combinatorica 18 (1998), 405-412.
FORMULA
T(n, k) = (1/2) * n!/(k!*(n-2*k)!).
EXAMPLE
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;
..
MAPLE
seq(seq(n!/(k!*(n-2*k)!)/2, k=1..floor(n/2)), n=1..20);
CROSSREFS
This is A059344/2 without column k=0.
Cf. A000407.
KEYWORD
nonn,tabf
AUTHOR
Peter C. Heinig (algorithms(AT)gmx.de), Feb 27 2008
STATUS
approved
page 1

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