A384651 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 A162661.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 7, 0, 1, 4, 9, 18, 33, 0, 1, 5, 14, 34, 84, 189, 0, 1, 6, 20, 56, 159, 472, 1249, 0, 1, 7, 27, 85, 265, 882, 3057, 9237, 0, 1, 8, 35, 122, 410, 1460, 5615, 22190, 74972, 0, 1, 9, 44, 168, 603, 2256, 9166, 40053, 177149, 659042, 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(2*n-j+k,j)/(2*n-j+k) * A(n-j,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,   33,   84,  159,  265,   410,   603, ...
  0,  189,  472,  882, 1460,  2256,  3330, ...
  0, 1249, 3057, 5615, 9166, 14015, 20540, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A162661.

Cf. A379598, A384652, A384653, A384654.

Sequence in context: A379598 A306704 A384626 * A091063 A384652 A384653

Adjacent sequences: A384648 A384649 A384650 * A384652 A384653 A384654

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 06 2025

Status

approved