A308484 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. log(1 + Sum_{j>=1} j^k * x^j/j!).

1, 1, 0, 1, 1, 0, 1, 3, -1, 0, 1, 7, -1, -2, 0, 1, 15, 5, -26, 9, 0, 1, 31, 35, -146, 29, 6, 0, 1, 63, 149, -650, -351, 756, -155, 0, 1, 127, 539, -2642, -5251, 9936, -1793, 232, 0, 1, 255, 1805, -10346, -46071, 83376, 51421, -45744, 3969, 0

Offset: 1

Author

Seiichi Manyama, May 30 2019

Keywords

• sign
• tabl

Examples

			Square array begins:
   1,    1,     1,     1,       1,        1, ...
   0,    1,     3,     7,      15,       31, ...
   0,   -1,    -1,     5,      35,      149, ...
   0,   -2,   -26,  -146,    -650,    -2642, ...
   0,    9,    29,  -351,   -5251,   -46071, ...
   0,    6,   756,  9936,   83376,   559656, ...
   0, -155, -1793, 51421, 1623439, 28735405, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 1..140, flattened

Crossrefs

Columns k=0..4 give A000007(n-1), A009306, A033464, A300452, A306325.

A(n,n) gives A320939.

Programs

Mathematica

T[n_, k_] := T[n, k] = n^k - Sum[Binomial[n-1,j] * j^k * T[n-j,k], {j,1,n-1}]; Table[T[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

Formula

A(n,k) = n^k - Sum_{j=1..n-1} binomial(n-1,j)*j^k*A(n-j,k).