A016309 -id:A016309 - cp's OEIS frontend

A071951 Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.

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

Offset: 1

N. J. A. Sloane, Jun 16 2002

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

			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

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.

Diagonals give A007290, A000079, A016129, A016309.
The column sequences are A000079 (powers of 2), A016129, A016309, A071952, A089274, A089277.
Cf. A080248, A089278, A089500, A160562.

[[(&+[(-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

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

Flatten[ Table[ Sum[(-1)^{r + j}(2r + 1)(r^2 + r)^n/((r + j + 1)!(j - r)!), {r, j}], {n, 10}, {j, n}]]

{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 */

[[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

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

A089278 Coefficient triangle for computation of column numbers of triangle A071951 (Legendre-Stirling).

Original entry on oeis.org

1, -1, 3, 1, -15, 24, -7, 405, -2268, 2500, 2, -405, 6048, -20000, 16875, -11, 7425, -266112, 2000000, -4640625, 3176523, 143, -312741, 25474176, -390000000, 1879453125, -3344878719, 1927561216, -143, 995085, -178319232, 5250000000, -46986328125, 163899057231, -236126248960

Offset: 1

Wolfdieter Lang, Nov 07 2003

The k-th column sequence A071951(n+k,k), n>=0, is sum(a(k,p)*(p*(p+1))^n,p=1..k)/A089500(k), k>=1.

			[1]; [ -1,3]; [1,-15,24]; [ -7,405,-2268,2500]; ...
Sequence A071951(n+3,3)= A016309(n)= [1,20,292,...] has a(n)=
(1*(1*2)^n - 15*(2*3)^n + 24*(3*4)^n)/10.

W. Lang, First 7 rows.

a(n, m)= A089500(n)*(((-1)^(m+n))*(2*m+1)*((m*(m+1))^n)/((m+n+1)!*(n-m)!)).

A089274 Fifth column of the Legendre-Stirling triangle A071951.

Original entry on oeis.org

1, 70, 3192, 121424, 4203824, 137922336, 4380918784, 136378114048, 4191383868672, 127754693361152, 3873052857829376, 117001609550671872, 3526270158211870720, 106112798944292282368, 3189880933574260359168

Offset: 0

Wolfdieter Lang, Nov 07 2003

This is the fifth member of the family A000079 (powers of 2), A016129, A016309, A071952, etc.

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).

Cf. A016129, A016309, A071951, A071952, A089278, A089500.

Cf. A000079 (powers of 2).

[(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

Table[2^(n-3)*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)/315, {n,0,30}] (* G. C. Greubel, Nov 10 2024 *)

def A089274(n): return 2^n*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)//2520
[A089274(n) for n in range(31)] # G. C. Greubel, Nov 10 2024

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]
E.g.f.: (1/2520)*(2*exp(2*x) - 405*exp(6*x) + 6048*exp(12*x) - 20000*exp(20*x) + 16875*exp(30*x)). - G. C. Greubel, Nov 10 2024

A089271 Third column (k=4) of array A078739(n,k) ((2,2)-generalized Stirling2).

Original entry on oeis.org

1, 38, 652, 9080, 116656, 1446368, 17636032, 213311360, 2569812736, 30898216448, 371141389312, 4455873443840, 53483541999616, 641880868118528, 7703040602324992, 92439308337643520, 1109288626710839296

Offset: 0

Wolfdieter Lang, Nov 07 2003

The numerator of the g.f. is the n=2 row polynomial of the triangle A089275.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
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 (20, -108, 144).

Cf. A089272, A071951 (Legendre-Stirling triangle).

[6*12^n-6*6^n+2^n: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

Table[6*12^n -6*6^n +2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)
LinearRecurrence[{20,-108,144},{1,38,652},20] (* Harvey P. Dale, Oct 22 2024 *)

for(n=0,30, print1(6*12^n -6*6^n +2^n, ", ")) \\ G. C. Greubel, Feb 07 2018

G.f.: (1+18*x)/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)).

a(n) = 6*12^n - 6*6^n + 2^n = d(n) + 18*d(n-1), n>=1, a(0)=1, with d(n) := A016309(n) = A071951(n+3, 3) = (24*12^n-15*6^n+2^n)/10.

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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.

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A089278 Coefficient triangle for computation of column numbers of triangle A071951 (Legendre-Stirling).

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A089274 Fifth column of the Legendre-Stirling triangle A071951.

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A089271 Third column (k=4) of array A078739(n,k) ((2,2)-generalized Stirling2).

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