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

Author

N. J. A. Sloane, Jun 19 2011

Keywords

• nonn
• tabl

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

Cf. A135921 (row sums), A191936.

Mirror of triangle A071951. - Omar E. Pol, Jan 10 2012

Programs

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

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