A362277 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/2)^j * binomial(n-j,j)/(n-j)!.

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -2, 1, 1, 1, -3, -8, 1, 6, 1, 1, 1, -4, -11, 10, 41, 16, 1, 1, 1, -5, -14, 25, 106, 31, -20, 1, 1, 1, -6, -17, 46, 201, -44, -461, -132, 1, 1, 1, -7, -20, 73, 326, -299, -1952, -895, 28, 1, 1, 1, -8, -23, 106, 481, -824, -5123, -1028, 6481, 1216, 1

Offset: 0

Author

Seiichi Manyama, Apr 13 2023

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,  1,  1,   1,    1,    1,     1, ...
  1,  1,  1,   1,    1,    1,     1, ...
  1,  0, -1,  -2,   -3,   -4,    -5, ...
  1, -2, -5,  -8,  -11,  -14,   -17, ...
  1, -2,  1,  10,   25,   46,    73, ...
  1,  6, 41, 106,  201,  326,   481, ...
  1, 16, 31, -44, -299, -824, -1709, ...
		

Links

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

Crossrefs

Columns k=0..6 give A000012, (-1)^n * A001464(n), A293604, A362278, A362176, A362279, A362177.

Main diagonal gives A362276.

T(n,2*n) gives A362282.

Cf. A359762, A362302.

Programs

PARI

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

Formula

E.g.f. of column k: exp(x - k*x^2/2).

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

T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j / (j! * (n-2*j)!).