A191936 Triangle read by rows of Legendre-Stirling numbers of the first kind.

1, 1, 0, 1, -2, 0, 1, -8, 12, 0, 1, -20, 108, -144, 0, 1, -40, 508, -2304, 2880, 0, 1, -70, 1708, -17544, 72000, -86400, 0, 1, -112, 4648, -89280, 808848, -3110400, 3628800, 0, 1, -168, 10920, -349568, 5808528, -48405888, 177811200, -203212800, 0

Offset: 1

Author

N. J. A. Sloane, Jun 19 2011

Keywords

• sign
• tabl

Comments

Apparently this is the mirror of triangle A129467. - Omar E. Pol, Jan 10 2012

Examples

			Triangle begins:
  1;
  1,    0;
  1,   -2,    0;
  1,   -8,   12,      0;
  1,  -20,  108,   -144,      0;
  1,  -40,  508,  -2304,   2880,        0;
  1,  -70, 1708, -17544,  72000,   -86400,       0;
  1, -112, 4648, -89280, 808848, -3110400, 3628800, 0;
  ...
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• G. E. Andrews, W. Gawronski and L. L. Littlejohn, The Legendre-Stirling Numbers
• G. E. Andrews et al., The Legendre-Stirling numbers, Discrete Math., 311 (2011), 1255-1272.
• J. Pan, Convolution Properties of the Generalized Stirling Numbers and the Jacobi-Stirling Numbers of the First Kind, Journal of Integer Sequences, 16 (2013), #13.9.2.

Crossrefs

Cf. A191935.

Programs

Mathematica

ps[n_, k_]:= ps[n, k]= If[k==n, 1, If[k==0, 0, ps[n-1, k-1] - n*(n-1)*ps[n-1, k]]];
Table[ps[n-1, n-k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)

Sage

@CachedFunction
def ps(n, k):
    if (k==n): return 1
    elif (k==0): return 0
    else: return ps(n-1, k-1) - n*(n-1)*ps(n-1, k)
flatten([[ps(n-1, n-k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jun 07 2021

Formula

T(n, k) = ps(n-1, n-k), where ps(n, k) = ps(n-1, k-1) - n*(n-1)*ps(n-1, k), ps(n, 0) = 0, and ps(n, n) = 1. - G. C. Greubel, Jun 07 2021

Extensions

More terms from Omar E. Pol, Jan 10 2012