A294046 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(1/(1-x)^k - 1).

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 13, 0, 1, 4, 21, 68, 73, 0, 1, 5, 36, 195, 580, 501, 0, 1, 6, 55, 424, 2241, 5912, 4051, 0, 1, 7, 78, 785, 6136, 30483, 69784, 37633, 0, 1, 8, 105, 1308, 13705, 104544, 476469, 933200, 394353, 0, 1, 9, 136, 2023, 26748

Offset

0,8

Links

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

Formula

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

Example

Square array A(n,k) begins:
   1,   1,    1,     1,      1, ...
   0,   1,    2,     3,      4, ...
   0,   3,   10,    21,     36, ...
   0,  13,   68,   195,    424, ...
   0,  73,  580,  2241,   6136, ...
   0, 501, 5912, 30483, 104544, ...

Mathematica

A[0, _] = 1; A[n_, k_] := k*(n-1)!*Sum[Binomial[j+k-1, k]*A[n-j, k]/(n-j)!, {j, 1, n}];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

Crossrefs

Columns k=0..5 give A000007, A000262, A136658, A202826, A294050, A294051.

Rows n=0..2 give A000012, A001477, A014105.

Main diagonal gives A294047.

Cf. A291709.

Sequence in context: A128888 A305401 A306100 * A320079 A349971 A340986

Adjacent sequences: A294043 A294044 A294045 * A294047 A294048 A294049

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 22 2017

Status

approved