A357293 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} Stirling2(n,k*j).

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 5, 0, 1, 0, 0, 3, 15, 0, 1, 0, 0, 1, 8, 52, 0, 1, 0, 0, 0, 6, 25, 203, 0, 1, 0, 0, 0, 1, 25, 97, 877, 0, 1, 0, 0, 0, 0, 10, 91, 434, 4140, 0, 1, 0, 0, 0, 0, 1, 65, 322, 2095, 21147, 0, 1, 0, 0, 0, 0, 0, 15, 350, 1232, 10707, 115975, 0, 1, 0, 0, 0, 0, 0, 1, 140, 1702, 5672, 58194, 678570, 0

Offset

0,9

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Eric Weisstein's World of Mathematics, Bell Polynomial.

Formula

For k > 0, e.g.f. of column k: Sum_{j>=0} (exp(x)-1)^(k*j)/(k*j)!.

For k > 0, T(n,k) = ( Sum_{j=0..k-1} Bell_n(w^j) )/k, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/k).

Example

Square array begins:
  1,   1,  1,  1,  1,  1, 1, ...
  0,   1,  0,  0,  0,  0, 0, ...
  0,   2,  1,  0,  0,  0, 0, ...
  0,   5,  3,  1,  0,  0, 0, ...
  0,  15,  8,  6,  1,  0, 0, ...
  0,  52, 25, 25, 10,  1, 0, ...
  0, 203, 97, 91, 65, 15, 1, ...

Prog

(PARI) T(n, k) = sum(j=0, n, stirling(n, k*j, 2));
(PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (exp(x+x*O(x^n))-1)^(k*j)/(k*j)!), n));
(PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Bell_poly(n, w^j)))/k);

Crossrefs

Columns k=0-3 give: A000007, A000110, A024430, A143815.

Cf. A357119.

Sequence in context: A112519 A276981 A230305 * A357119 A357883 A091728

Adjacent sequences: A357290 A357291 A357292 * A357294 A357295 A357296

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 17 2022

Status

approved