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A191936 Triangle read by rows of Legendre-Stirling numbers of the first kind. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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

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.

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

EXAMPLE

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;

  ...

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 *)

PROG

(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

CROSSREFS

Cf. A191935.

Sequence in context: A055141 A055140 A335330 * A327090 A021836 A255306

Adjacent sequences:  A191933 A191934 A191935 * A191937 A191938 A191939

KEYWORD

sign,tabl

AUTHOR

N. J. A. Sloane, Jun 19 2011

EXTENSIONS

More terms from Omar E. Pol, Jan 10 2012

STATUS

approved

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Last modified January 21 16:03 EST 2022. Contains 350479 sequences. (Running on oeis4.)