A380841 Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 21, 10, 3, 1, 0, 148, 66, 18, 4, 1, 0, 1305, 560, 141, 28, 5, 1, 0, 13806, 5770, 1380, 252, 40, 6, 1, 0, 170401, 69852, 16095, 2776, 405, 54, 7, 1, 0, 2403640, 970886, 217458, 35940, 4940, 606, 70, 8, 1, 0, 38143377, 15228880, 3335745, 533304, 70045, 8088, 861, 88, 9, 1

Offset: 0

Author

Stefano Spezia, Feb 05 2025

Keywords

• nonn
• tabl

Examples

			Array begins as:
  1,    1,    1,     1,     1,     1,      1, ...
  0,    1,    2,     3,     4,     5,      6, ...
  0,    4,   10,    18,    28,    40,     54, ...
  0,   21,   66,   141,   252,   405,    606, ...
  0,  148,  560,  1380,  2776,  4940,   8088, ...
  0, 1305, 5770, 16095, 35940, 70045, 124350, ...
  ...
		

Crossrefs

Cf. A380843 (antidiagonal sums).

Columns k=0..4 give A000007, A006153, A377529, A377530, A379993.

Rows n=0..2 give A000012, A001477, A028552.

Main diagonal gives A380842.

A(n,n+1) gives A213643(n+1).

Programs

Mathematica

A[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

Formula

A(n,k) = n! * Sum_{j=0..n} j^(n-j) * binomial(j+k-1,j)/(n-j)!. - Seiichi Manyama, Feb 06 2025