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%I #32 Dec 18 2023 04:02:29
%S 1,1,2,1,8,4,1,20,52,8,1,40,292,320,16,1,70,1092,3824,1936,32,1,112,
%T 3192,25664,47824,11648,64,1,168,7896,121424,561104,585536,69952,128,
%U 1,240,17304,453056,4203824,11807616,7096384,419840,256,1,330,34584,1422080,23232176,137922336,243248704,85576448,2519296,512
%N Triangle read by rows of Legendre-Stirling numbers of the second kind.
%H G. C. Greubel, <a href="/A191935/b191935.txt">Rows n = 1..50 of the triangle, flattened</a>
%H G. E. Andrews, W. Gawronski and L. L. Littlejohn, <a href="https://georgeandrews1.github.io/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.
%F From _G. C. Greubel_, Jun 06 2021: (Start)
%F T(n, k) = Ps(n, n-k+1), where Ps(n, k) = Sum_{j=0..k} (-1)^(j+k)*(2*j+1)*j^n*(1 + j)^n/((j+k+1)!*(k-j)!).
%F Sum_{k=1..n} T(n, k) = A135921(n). (End)
%e Triangle begins:
%e 1;
%e 1 2;
%e 1 8 4;
%e 1 20 52 8;
%e 1 40 292 320 16;
%e 1 70 1092 3824 1936 32;
%e 1 112 3192 25664 47824 11648 64;
%e 1 168 7896 121424 561104 585536 69952 128;
%e ...
%t Ps[n_, k_]:= Sum[(-1)^(j+k)*(2*j+1)*j^n*(1+j)^n/((j+k+1)!*(k-j)!), {j,0,k}];
%t Table[Ps[n, n-k+1], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Jun 06 2021 *)
%o (PARI) T071951(n, k) = sum(i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! );
%o for (n=1, 10, for (k=1, n, print1(T071951(n,n-k+1), ", ")); print); \\ _Michel Marcus_, Nov 24 2019
%o (Sage)
%o def Ps(n,k): return sum( (-1)^(j+k)*(2*j+1)*j^n*(1+j)^n/(factorial(j+k+1) * factorial(k-j)) for j in (0..k) )
%o flatten([[Ps(n,n-k+1) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jun 06 2021
%Y Cf. A135921 (row sums), A191936.
%Y Mirror of triangle A071951. - _Omar E. Pol_, Jan 10 2012
%K nonn,tabl
%O 1,3
%A _N. J. A. Sloane_, Jun 19 2011
%E More terms from _Omar E. Pol_, Jan 10 2012
%E More terms from _Michel Marcus_, Nov 24 2019
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