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 A191935 Triangle read by rows of Legendre-Stirling numbers of the second kind. 2
 1, 1, 2, 1, 8, 4, 1, 20, 52, 8, 1, 40, 292, 320, 16, 1, 70, 1092, 3824, 1936, 32, 1, 112, 3192, 25664, 47824, 11648, 64, 1, 168, 7896, 121424, 561104, 585536, 69952, 128, 1, 240, 17304, 453056, 4203824, 11807616, 7096384, 419840, 256, 1, 330, 34584, 1422080, 23232176, 137922336, 243248704, 85576448, 2519296, 512 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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. FORMULA From G. C. Greubel, Jun 06 2021: (Start) 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)!). Sum_{k=1..n} T(n, k) = A135921(n). (End) EXAMPLE Triangle begins:   1;   1   2;   1   8    4;   1  20   52      8;   1  40  292    320     16;   1  70 1092   3824   1936     32;   1 112 3192  25664  47824  11648    64;   1 168 7896 121424 561104 585536 69952 128;   ... MATHEMATICA Ps[n_, k_]:= Sum[(-1)^(j+k)*(2*j+1)*j^n*(1+j)^n/((j+k+1)!*(k-j)!), {j, 0, k}]; Table[Ps[n, n-k+1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jun 06 2021 *) PROG (PARI) T071951(n, k) = sum(i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! ); for (n=1, 10, for (k=1, n, print1(T071951(n, n-k+1), ", ")); print); \\ Michel Marcus, Nov 24 2019 (Sage) 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) ) flatten([[Ps(n, n-k+1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jun 06 2021 CROSSREFS Cf. A135921 (row sums), A191936. Mirror of triangle A071951. - Omar E. Pol, Jan 10 2012 Sequence in context: A234014 A208931 A133214 * A156365 A142075 A110107 Adjacent sequences:  A191932 A191933 A191934 * A191936 A191937 A191938 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Jun 19 2011 EXTENSIONS More terms from Omar E. Pol, Jan 10 2012 More terms from Michel Marcus, Nov 24 2019 STATUS approved

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Last modified June 21 19:59 EDT 2021. Contains 345365 sequences. (Running on oeis4.)