A355652 -id:A355652 - cp's OEIS frontend

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

Seiichi Manyama, Jul 13 2022

			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, ...

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

Cf. A355609, A355652.

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

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)!.

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)).

Original entry on oeis.org

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

Seiichi Manyama, Jul 13 2022

			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, ...

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

Cf. A351703, A355652.

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

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)!).

cp's OEIS Frontend

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)).

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