A357868 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j).

1, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 0, 2, 13, 0, 1, 0, 0, 6, 75, 0, 1, 0, 0, 6, 38, 541, 0, 1, 0, 0, 0, 36, 270, 4683, 0, 1, 0, 0, 0, 24, 150, 2342, 47293, 0, 1, 0, 0, 0, 0, 240, 1260, 23646, 545835, 0, 1, 0, 0, 0, 0, 120, 1560, 16926, 272918, 7087261, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 197316, 3543630, 102247563, 0

Offset

0,9

Links

Table of n, a(n) for n=0..77.

Formula

For k > 0, e.g.f. of column k: 1/(1 - (exp(x) - 1)^k).

T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * Stirling2(j,k) * T(n-j,k).

Example

Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   0,   0,   0,   0, ...
  0,   3,   2,   0,   0,   0, ...
  0,  13,   6,   6,   0,   0, ...
  0,  75,  38,  36,  24,   0, ...
  0, 541, 270, 150, 240, 120, ...

Prog

(PARI) T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2));
(PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(1/(1-(exp(x+x*O(x^n))-1)^k), n));

Crossrefs

Columns k=0-4 give: A000007, A000670, A052841, A353774, A353775.

Cf. A324162, A357293, A357869, A357881.

Sequence in context: A351097 A155522 A007524 * A357881 A204689 A109718

Adjacent sequences: A357865 A357866 A357867 * A357869 A357870 A357871

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 17 2022

Status

approved