A362302 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (-k/6)^j * binomial(n-2*j,j)/(n-2*j)!.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -1, -3, 1, 1, 1, 1, -2, -7, -9, 1, 1, 1, 1, -3, -11, -19, -9, 1, 1, 1, 1, -4, -15, -29, 1, 36, 1, 1, 1, 1, -5, -19, -39, 31, 211, 225, 1, 1, 1, 1, -6, -23, -49, 81, 526, 1009, 477, 1, 1, 1, 1, -7, -27, -59, 151, 981, 2353, 953, -819, 1

Offset: 0

Author

Seiichi Manyama, Apr 15 2023

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,  1,   1,   1,   1,   1,   1, ...
  1,  1,   1,   1,   1,   1,   1, ...
  1,  1,   1,   1,   1,   1,   1, ...
  1,  0,  -1,  -2,  -3,  -4,  -5, ...
  1, -3,  -7, -11, -15, -19, -23, ...
  1, -9, -19, -29, -39, -49, -59, ...
  1, -9,   1,  31,  81, 151, 241, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

Columns k=0..2 give A000012, A351929, A362309.

Main diagonal gives A362303.

T(n,2*n) gives A362304.

T(n,6*n) gives A362305.

Cf. A362043, A362277.

Programs

PARI

T(n, k) = n!*sum(j=0, n\3, (-k/6)^j/(j!*(n-3*j)!));

Formula

E.g.f. of column k: exp(x - k*x^3/6).

T(n,k) = T(n-1,k) - k * binomial(n-1,2) * T(n-3,k) for n > 2.

T(n,k) = n! * Sum_{j=0..floor(n/3)} (-k/6)^j / (j! * (n-3*j)!).