A384626 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A143426.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 7, 0, 1, 4, 9, 18, 32, 0, 1, 5, 14, 34, 82, 175, 0, 1, 6, 20, 56, 156, 442, 1086, 0, 1, 7, 27, 85, 261, 834, 2699, 7429, 0, 1, 8, 35, 122, 405, 1392, 5027, 18178, 54994, 0, 1, 9, 44, 168, 597, 2166, 8310, 33387, 132664, 435120, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(n-j+k,j)/(n-j+k) * A(n-j,2*j).

Example

Square array begins:
  1,    1,    1,    1,    1,     1,     1, ...
  0,    1,    2,    3,    4,     5,     6, ...
  0,    2,    5,    9,   14,    20,    27, ...
  0,    7,   18,   34,   56,    85,   122, ...
  0,   32,   82,  156,  261,   405,   597, ...
  0,  175,  442,  834, 1392,  2166,  3216, ...
  0, 1086, 2699, 5027, 8310, 12850, 19022, ...

Prog

(PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 2*j)));

Crossrefs

Columns k=0..1 give A000007, A143426.

Sequence in context: A378291 A379598 A306704 * A384651 A091063 A384652

Adjacent sequences: A384623 A384624 A384625 * A384627 A384628 A384629

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 05 2025

Status

approved