A293991 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j=1..k+1} binomial(k,j-1)*x^j/j).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 9, 10, 1, 1, 1, 5, 16, 33, 26, 1, 1, 1, 6, 25, 76, 141, 76, 1, 1, 1, 7, 36, 145, 436, 651, 232, 1, 1, 1, 8, 49, 246, 1025, 2776, 3333, 764, 1, 1, 1, 9, 64, 385, 2046, 8245, 19384, 18369, 2620, 1, 1, 1, 10, 81

Offset: 0

Author

Seiichi Manyama, Oct 21 2017

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
   1,  1,   1,   1,    1, ...
   1,  1,   1,   1,    1, ...
   1,  2,   3,   4,    5, ...
   1,  4,   9,  16,   25, ...
   1, 10,  33,  76,  145, ...
   1, 26, 141, 436, 1025, ...
		

Links

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

Crossrefs

Columns k=0..5 give A000012, A000085, A049425, A049426, A049427, A049428.

Rows n=0-1 give A000012.

Main diagonal gives A294003.

Cf. A291709.

Programs

Mathematica

A[0, ] = 1; A[n, k_] := (n-1)!*Sum[Binomial[k, j-1]*A[n-j, k]/(n-j)!, {j, 1, Min[k+1, n]}];
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

Formula

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

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k+1,n)} binomial(k,j-1)*A(n-j,k)/(n-j)! for n > 0.