A191935 Triangle read by rows of Legendre-Stirling numbers of the second kind.
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
Offset: 1
Examples
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; ...
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.
Crossrefs
Mirror of triangle A071951. - Omar E. Pol, Jan 10 2012
Programs
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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 *) -
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
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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
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)
Extensions
More terms from Omar E. Pol, Jan 10 2012
More terms from Michel Marcus, Nov 24 2019