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%I #29 Jun 07 2021 04:38:54
%S 1,1,0,1,-2,0,1,-8,12,0,1,-20,108,-144,0,1,-40,508,-2304,2880,0,1,-70,
%T 1708,-17544,72000,-86400,0,1,-112,4648,-89280,808848,-3110400,
%U 3628800,0,1,-168,10920,-349568,5808528,-48405888,177811200,-203212800,0
%N Triangle read by rows of Legendre-Stirling numbers of the first kind.
%C Apparently this is the mirror of triangle A129467. - _Omar E. Pol_, Jan 10 2012
%H G. C. Greubel, <a href="/A191936/b191936.txt">Rows n = 1..50 of the triangle, flattened</a>
%H G. E. Andrews, W. Gawronski and L. L. Littlejohn, <a href="http://www.math.psu.edu/andrews/pdf/283.pdf">The Legendre-Stirling Numbers</a>
%H G. E. Andrews et al., <a href="https://doi.org/10.1016/j.disc.2011.02.028">The Legendre-Stirling numbers</a>, Discrete Math., 311 (2011), 1255-1272.
%H J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pan/pan21.html">Convolution Properties of the Generalized Stirling Numbers and the Jacobi-Stirling Numbers of the First Kind</a>, Journal of Integer Sequences, 16 (2013), #13.9.2.
%F 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
%e Triangle begins:
%e 1;
%e 1, 0;
%e 1, -2, 0;
%e 1, -8, 12, 0;
%e 1, -20, 108, -144, 0;
%e 1, -40, 508, -2304, 2880, 0;
%e 1, -70, 1708, -17544, 72000, -86400, 0;
%e 1, -112, 4648, -89280, 808848, -3110400, 3628800, 0;
%e ...
%t 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]]];
%t Table[ps[n-1, n-k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jun 07 2021 *)
%o (Sage)
%o @CachedFunction
%o def ps(n, k):
%o if (k==n): return 1
%o elif (k==0): return 0
%o else: return ps(n-1, k-1) - n*(n-1)*ps(n-1, k)
%o flatten([[ps(n-1, n-k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jun 07 2021
%Y Cf. A191935.
%K sign,tabl
%O 1,5
%A _N. J. A. Sloane_, Jun 19 2011
%E More terms from _Omar E. Pol_, Jan 10 2012
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