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Search: a071952 -id:a071952
Displaying 1-5 of 5 results found. page 1
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A071951 Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows. +10
23
1, 2, 1, 4, 8, 1, 8, 52, 20, 1, 16, 320, 292, 40, 1, 32, 1936, 3824, 1092, 70, 1, 64, 11648, 47824, 25664, 3192, 112, 1, 128, 69952, 585536, 561104, 121424, 7896, 168, 1, 256, 419840, 7096384, 11807616, 4203824, 453056, 17304, 240, 1, 512, 2519296, 85576448, 243248704, 137922336, 23232176, 1422080, 34584, 330, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Removing a factor of 2^m from the m-th subdiagonal (the main diagonal corresponds to m = 0) gives the triangle A080248. - Peter Bala, Oct 15 2023
LINKS
Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
G. E. Andrews, W. Gawronski and L. L. Littlejohn, The Legendre-Stirling Numbers, Discrete Mathematics, Volume 311, Issue 14, 28 July 2011, Pages 1255-1272.
M. W. Coffey, M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
José L. Cereceda, A refinement of Lang's formula for the sum of powers of integers, arXiv:2301.02141 [math.NT], 2023.
E. S. Egge, Legendre-Stirling permutations, Eur. J. Combin. 13 (2010) 1735-1750.
W. N. Everitt, L. L. Littlejohn and R. Wellman, Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math. 148, 2002, 213-238.
H. Li, T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, Theorem 43.
L. L. Littlejohn and R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181(2), 2002, 280-339.
FORMULA
T(n, j) = Sum_{r=1..j} (-1)^(r+j)*(2*r+1)*(r^2+r)^n/((r+j+1)!*(j-r)!).
G.f. for j-th column (without leading zeros): 1/Product_{r=1..j} (1 - r*(r+1)*x), j >= 1. From eq.(4.5) of the Everitt et al. paper.
A135921(n+1) = row sums. - Michael Somos, Feb 25 2012
Sum_{n=j..m} binomial(m,n)*T(n,j)*4^(n-j) = A160562(m,j) for 1 <= j <= m. - Werner Schulte, Dec 03 2015
EXAMPLE
The triangle begins:
n\j 1 2 3 4 5 6 7 8 9 ...
1: 1
2: 2 1
3: 4 8 1
4: 8 52 20 1
5: 16 320 292 40 1
6: 32 1936 3824 1092 70 1
7: 64 11648 47824 25664 3192 112 1
8: 128 69952 585536 561104 121424 7896 168 1
9: 256 419840 7096384 11807616 4203824 453056 17304 240 1
...
Row 10: 512 2519296 85576448 243248704 137922336 23232176 1422080 34584 330 1. Reformatted by Wolfdieter Lang, Apr 10 2013
MAPLE
N:= 20: # to get the first N rows, flattened
for j from 1 to N do S[j]:= series(x^j/mul(1-r*(r+1)*x, r=1..j), x, N+1) od:
seq(seq(coeff(S[j], x, i), j=1..i), i=1..N); # Robert Israel, Dec 03 2015
# alternative
A071951 := proc(n, k)
option remember;
if k =0 then
if n = 0 then
1;
else
0;
end if;
elif n = 0 then
if k =0 then
1;
else
0;
end if;
else
procname(n-1, k-1)+k*(k+1)*procname(n-1, k) ;
end if;
end proc: # R. J. Mathar, Jun 30 2018
MATHEMATICA
Flatten[ Table[ Sum[(-1)^{r + j}(2r + 1)(r^2 + r)^n/((r + j + 1)!(j - r)!), {r, j}], {n, 10}, {j, n}]]
PROG
(PARI) {T(n, k) = sum( i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! )} /* Michael Somos, Feb 25 2012 */
(Magma) [[(&+[(-1)^(r+j)*(2*r+1)*(r^2+r)^n/(Factorial(r+j+1)*Factorial(j-r)): r in [1..j]]): j in [1..n]]: n in [1..12]]; // G. C. Greubel, Mar 16 2019
(Sage) [[sum( (-1)^(r+j)*(2*r+1)*(r^2+r)^n/(factorial(r+j+1)*factorial(j-r)) for r in (1..j)) for j in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 16 2019
CROSSREFS
Diagonals give A007290, A000079, A016129, A016309.
The column sequences are A000079 (powers of 2), A016129, A016309, A071952, A089274, A089277.
Cf. A080248, A089278, A089500, A160562.
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jun 16 2002
STATUS
approved
A089272 Fourth column (k=5) of array A078739(n,k) ((2,2)- generalized Stirling2) divided by 12. +10
3
1, 48, 1412, 34400, 766416, 16296448, 337709632, 6896540160, 139644851456, 2813500878848, 56517475402752, 1133320271749120, 22702062218039296, 454469171469877248, 9094518828981174272, 181952003020274401280 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The numerator of the g.f. is the n=2 row polynomial of the triangle A089276.
REFERENCES
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003), 198-205.
LINKS
Table of n, a(n) for n=0..15.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Index entries for linear recurrences with constant coefficients, signature (40, -508, 2304, -2880).
FORMULA
G.f. (1+8*x)/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)).
a(n)= (3500*20^n - 3780*12^n + 945*6^n - 35*2^n)/630 = d(n) + 8*d(n-1), with d(n) := A071952(n+4)= (2500*20^n - 2268*12^n + 405*6^n - 7*2^n)/630, n>=1.
MATHEMATICA
LinearRecurrence[{40, -508, 2304, -2880}, {1, 48, 1412, 34400}, 16] (* Jean-François Alcover, Feb 28 2020 *) (* Jean-François Alcover, Feb 28 2020 *)
CROSSREFS
Cf. A071952, A089271, A089273, A071951 (Legendre-Stirling triangle).
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 07 2003
STATUS
approved
A089274 Fifth column of the Legendre-Stirling triangle A071951. +10
3
1, 70, 3192, 121424, 4203824, 137922336, 4380918784, 136378114048, 4191383868672, 127754693361152, 3873052857829376, 117001609550671872, 3526270158211870720, 106112798944292282368, 3189880933574260359168 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
This is the fifth member of the family A000079 (powers of 2), A016129, A016309, A071952, etc.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (70, -1708, 17544, -72000, 86400).
FORMULA
G.f.: 1/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)*(1-6*5*x)).
a(n) = (16875*(6*5)^n -20000*(5*4)^n +6048*(4*3)^n -405*(3*2)^n +2*(2*1)^n)/2520.
a(n) = A071951(n+5, 5), n>=0.
a(n) = det(|ps(i+5,j+4)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). [Mircea Merca, Apr 06 2013]
PROG
(Magma) [(16875*(6*5)^n - 20000*(5*4)^n + 6048*(4*3)^n - 405*(3*2)^n + 2*(2*1)^n)/2520: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011
CROSSREFS
Cf. A089278, A089500.
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 07 2003
STATUS
approved
A093050 Exponent of 2 in (3^n-3)*2^(n-1). +10
2
0, 0, 3, 2, 6, 4, 7, 6, 11, 8, 11, 10, 14, 12, 15, 14, 20, 16, 19, 18, 22, 20, 23, 22, 27, 24, 27, 26, 30, 28, 31, 30, 37, 32, 35, 34, 38, 36, 39, 38, 43, 40, 43, 42, 46, 44, 47, 46, 52, 48, 51, 50, 54, 52, 55, 54, 59, 56, 59, 58, 62, 60, 63, 62, 70, 64, 67, 66, 70 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..68.
FORMULA
Recurrence: a(2n) = a(n) + [(n+1)/2] + 1, a(2n+1) = 2n.
G.f.: sum(k>=0, t^2(3+2t+2t^3-t^4)/[(1+t^2)(1-t^2)^2], t=x^2^k).
PROG
(PARI) a(n)=if(n<1, 0, if(n%2==0, a(n/2)+2*floor((n+2)/4)+1, n-1))
CROSSREFS
Cf. A093051, A093052.
a(n) is the exponent of 2 in A016129(n-1), A024281(n), A024287(n), A066406(n)/2, A071952(n+3).
KEYWORD
nonn,changed
AUTHOR
Ralf Stephan, Mar 16 2004
STATUS
approved
A071953 Diagonal T(n,n-2) of triangle in A071951. +10
1
4, 52, 292, 1092, 3192, 7896, 17304, 34584, 64284, 112684, 188188, 301756, 467376, 702576, 1028976, 1472880, 2065908, 2845668, 3856468, 5150068, 6786472, 8834760, 11373960, 14493960, 18296460, 22895964, 28420812, 35014252 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 3..5000
W. N. Everitt, L. L. Littlejohn and R. Wellman, Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math. 148, 2002, 213-238.
L. L. Littlejohn and R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181(2), 2002, 280-339.
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).
FORMULA
a(n) = (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90.
a(0)=4, a(1)=52, a(2)=292, a(3)=1092, a(4)=3192, a(5)=7896, a(6)=17304, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Jul 03 2011
G.f.: 4*(3*x*(x+2)+1)/(1-x)^7. - Harvey P. Dale, Jul 03 2011
E.g.f.: x^3*(60 + 135*x + 54*x^2 + 5*x^3)*exp(x)/90. - G. C. Greubel, Mar 16 2019
MATHEMATICA
Flatten[ Table[ Sum[(-1)^{r + n - 2}(2r + 1)(r^2 + r)^n/((r + n - 1)!(n - 2 - r)!), {r, 1, n - 2}], {n, 3, 34}]]
Table[(n-2)(n-1)n(n+1)(5n^2-11n+3)/90, {n, 3, 30}] (* or *) LinearRecurrence[ {7, -21, 35, -35, 21, -7, 1}, {4, 52, 292, 1092, 3192, 7896, 17304}, 30] (* Harvey P. Dale, Jul 03 2011 *)
PROG
(PARI) {a(n) = (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90}; \\ G. C. Greubel, Mar 16 2019
(Magma) [(n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90: n in [3..30]]; // G. C. Greubel, Mar 16 2019
(Sage) [(n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90 for n in (3..30)] # G. C. Greubel, Mar 16 2019
(GAP) List([3..30], n-> (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90); # G. C. Greubel, Mar 16 2019
CROSSREFS
Cf. A071951, A071952.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 16 2002
EXTENSIONS
More terms from Robert G. Wilson v, Jun 19 2002
STATUS
approved
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Last modified September 8 15:19 EDT 2024. Contains 375753 sequences. (Running on oeis4.)