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

1,3

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: A367994 A208931 A133214 * A156365 A142075 A366173

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