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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
OFFSET
1,5
COMMENTS
Apparently this is the mirror of triangle A129467. - Omar E. Pol, Jan 10 2012
LINKS
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.
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
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Jun 19 2011
EXTENSIONS
More terms from Omar E. Pol, Jan 10 2012
STATUS
approved