A383341 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * binomial(j+k,j)/(n-j)!.

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 11, 24, 1, 1, 5, 16, 53, 120, 1, 1, 6, 21, 88, 309, 720, 1, 1, 7, 26, 129, 568, 2119, 5040, 1, 1, 8, 31, 176, 897, 4288, 16687, 40320, 1, 1, 9, 36, 229, 1296, 7317, 36832, 148329, 362880, 1, 1, 10, 41, 288, 1765, 11296, 67365, 354688, 1468457, 3628800

Offset: 0

Author

Seiichi Manyama, Apr 24 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
    1,    1,    1,    1,     1,     1,     1, ...
    1,    1,    1,    1,     1,     1,     1, ...
    2,    3,    4,    5,     6,     7,     8, ...
    6,   11,   16,   21,    26,    31,    36, ...
   24,   53,   88,  129,   176,   229,   288, ...
  120,  309,  568,  897,  1296,  1765,  2304, ...
  720, 2119, 4288, 7317, 11296, 16315, 22464, ...
		

Crossrefs

Columns k=0..4 give A000142, A000255, A052124, A383378, A383383.

Main diagonal gives A383379.

Cf. A295181.

Programs

PARI

a(n,k) = n!*sum(j=0, n, (-k)^(n-j)*binomial(j+k, j)/(n-j)!);

Formula

E.g.f. of column k: exp(-k*x) / (1-x)^(k+1).

A(0,k) = A(1,k) = 1; A(n,k) = n*A(n-1,k) + k*(n-1)*A(n-2,k).