A362834 -id:A362834 - cp's OEIS frontend

A361652 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^j * Stirling2(n-j,j)/(n-j)!.

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 6, 16, 0, 1, 0, 8, 9, 56, 65, 0, 1, 0, 10, 12, 120, 250, 336, 0, 1, 0, 12, 15, 208, 555, 1812, 1897, 0, 1, 0, 14, 18, 320, 980, 5148, 12614, 11824, 0, 1, 0, 16, 21, 456, 1525, 11064, 39711, 101040, 80145, 0

Offset: 0

Seiichi Manyama, May 05 2023

			Square array begins:
  1,  1,   1,   1,   1,    1, ...
  0,  0,   0,   0,   0,    0, ...
  0,  2,   4,   6,   8,   10, ...
  0,  3,   6,   9,  12,   15, ...
  0, 16,  56, 120, 208,  320, ...
  0, 65, 250, 555, 980, 1525, ...

Columns k=0..3 give: A000007, A052506, A351733, A351734.
Main diagonal gives (-1)^n * A290158(n).
Cf. A362834.

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

E.g.f. of column k: exp(k * x * (exp(x) - 1)).

A362837 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling1(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 20, 0, 1, 0, 8, 27, 112, 90, 0, 1, 0, 10, 48, 324, 960, 594, 0, 1, 0, 12, 75, 704, 4050, 10848, 4200, 0, 1, 0, 14, 108, 1300, 11520, 64962, 141120, 34544, 0, 1, 0, 16, 147, 2160, 26250, 239616, 1224720, 2122496, 316008, 0

Offset: 0

Seiichi Manyama, May 05 2023

			Square array begins:
  1,  1,   1,    1,     1,     1, ...
  0,  0,   0,    0,     0,     0, ...
  0,  2,   4,    6,     8,    10, ...
  0,  3,  12,   27,    48,    75, ...
  0, 20, 112,  324,   704,  1300, ...
  0, 90, 960, 4050, 11520, 26250, ...

Columns k=0..3 give: A000007, A066166, A053491, A351735.
Main diagonal gives A362838.
Cf. A362834.

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

E.g.f. of column k: 1/(1 - k * x)^x.

cp's OEIS Frontend

A361652 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^j * Stirling2(n-j,j)/(n-j)!.

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