A253667 Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](exp(-x) *sum(j=0..n, C(n,j)*x^j)), n>=0, k>=0.

1, 1, -1, 1, 0, 1, 1, 1, -1, -1, 1, 2, -1, 2, 1, 1, 3, 1, -1, -3, -1, 1, 4, 5, -4, 5, 4, 1, 1, 5, 11, -1, 1, -11, -5, -1, 1, 6, 19, 14, -15, 14, 19, 6, 1, 1, 7, 29, 47, -19, 19, -47, -29, -7, -1, 1, 8, 41, 104, 37, -56, 37, 104, 41, 8, 1

Offset: 0

Author

Peter Luschny, Jan 18 2015

Keywords

• sign
• tabl

Examples

			Square array starts:
[n\k][0   1   2   3    4     5     6]
[0]   1, -1,  1, -1,   1,   -1,    1, ...
[1]   1,  0, -1,  2,  -3,    4,   -5, ...
[2]   1,  1, -1, -1,   5,  -11,   19, ...
[3]   1,  2,  1, -4,   1,   14,  -47, ...
[4]   1,  3,  5, -1, -15,   19,   37, ...
[5]   1,  4, 11, 14, -19,  -56,  151, ...
[6]   1,  5, 19, 47,  37, -151, -185, ...
The first few rows as a triangle:
1,
1, -1,
1,  0,  1,
1,  1, -1, -1,
1,  2, -1,  2,  1,
1,  3,  1, -1, -3, -1,
1,  4,  5, -4,  5,  4, 1.
		

Crossrefs

Cf. A009940.

Programs

Maple

T := (n,k) -> k!*coeff(series(exp(-x)*add(binomial(n,j)*x^j, j=0..n), x, k+1), x, k): for n from 0 to 6 do lprint(seq(T(n,k),k=0..6)) od;

Formula

T(n,n) = A009940(n).