A355666 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k/k! * (exp(x) - 1)).

1, 1, 1, 1, 0, 3, 1, 0, 2, 13, 1, 0, 0, 3, 75, 1, 0, 0, 3, 28, 541, 1, 0, 0, 0, 6, 125, 4683, 1, 0, 0, 0, 4, 10, 1146, 47293, 1, 0, 0, 0, 0, 10, 195, 8827, 545835, 1, 0, 0, 0, 0, 5, 20, 1281, 94200, 7087261, 1, 0, 0, 0, 0, 0, 15, 35, 5908, 1007001, 102247563, 1, 0, 0, 0, 0, 0, 6, 35, 1176, 68076, 12814390, 1622632573

Offset

0,6

Links

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

Formula

T(0,k) = 1 and T(n,k) = binomial(n,k) * Sum_{j=k+1..n} binomial(n-k,j-k) * T(n-j,k) for n > 0.

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).

Example

Square array begins:
     1,    1,   1,  1,  1, 1, 1, ...
     1,    0,   0,  0,  0, 0, 0, ...
     3,    2,   0,  0,  0, 0, 0, ...
    13,    3,   3,  0,  0, 0, 0, ...
    75,   28,   6,  4,  0, 0, 0, ...
   541,  125,  10, 10,  5, 0, 0, ...
  4683, 1146, 195, 20, 15, 6, 0, ...

Prog

(PARI) T(n, k) = n!*sum(j=0, n\(k+1), j!*stirling(n-k*j, j, 2)/(k!^j*(n-k*j)!));

Crossrefs

Columns k=0..3 give A000670, A052848, A353998, A353999.

Cf. A351703, A355652.

Sequence in context: A194582 A357438 A324173 * A293134 A293053 A355652

Adjacent sequences: A355663 A355664 A355665 * A355667 A355668 A355669

Keyword

nonn,tabl

Author

Seiichi Manyama, Jul 13 2022

Status

approved