A355665 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 * log(1 - x)).

1, 1, 1, 1, 0, 3, 1, 0, 2, 14, 1, 0, 0, 3, 88, 1, 0, 0, 6, 32, 694, 1, 0, 0, 0, 12, 150, 6578, 1, 0, 0, 0, 24, 40, 1524, 72792, 1, 0, 0, 0, 0, 60, 900, 12600, 920904, 1, 0, 0, 0, 0, 120, 240, 6048, 147328, 13109088, 1, 0, 0, 0, 0, 0, 360, 1260, 43680, 1705536, 207360912

Offset

0,6

Links

Table of n, a(n) for n=0..65.

Formula

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

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * |Stirling1(n-k*j,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, ...
    14,    3,   6,   0,   0,   0, 0, ...
    88,   32,  12,  24,   0,   0, 0, ...
   694,  150,  40,  60, 120,   0, 0, ...
  6578, 1524, 900, 240, 360, 720, 0, ...

Prog

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

Crossrefs

Columns k=0..3 give A007840, A052830, A351503, A351504.

Cf. A355609, A355652.

Sequence in context: A293134 A293053 A355652 * A144108 A163972 A068464

Adjacent sequences: A355662 A355663 A355664 * A355666 A355667 A355668

Keyword

nonn,tabl

Author

Seiichi Manyama, Jul 13 2022

Status

approved