A362043 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, 2, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 4, 9, 11, 1, 1, 1, 1, 5, 13, 21, 31, 1, 1, 1, 1, 6, 17, 31, 81, 106, 1, 1, 1, 1, 7, 21, 41, 151, 351, 337, 1, 1, 1, 1, 8, 25, 51, 241, 736, 1233, 1205, 1, 1, 1, 1, 9, 29, 61, 351, 1261, 2689, 5769, 5021, 1

Offset: 0

Author

Seiichi Manyama, Apr 15 2023

Keywords

• nonn
• 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,  2,  3,   4,   5,   6,   7, ...
  1,  5,  9,  13,  17,  21,  25, ...
  1, 11, 21,  31,  41,  51,  61, ...
  1, 31, 81, 151, 241, 351, 481, ...
		

Links

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

Crossrefs

Columns k=0..2 give A000012, A190865, A001470.

Main diagonal gives A362173.

T(n,2*n) gives A362300.

T(n,6*n) gives A362301.

Cf. A359762, A362302.

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)!).