A384580 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 A143500.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 8, 10, 0, 1, 5, 10, 16, 27, 46, 0, 1, 6, 15, 28, 54, 118, 244, 0, 1, 7, 21, 45, 95, 228, 609, 1481, 0, 1, 8, 28, 68, 155, 392, 1144, 3602, 10020, 0, 1, 9, 36, 98, 240, 631, 1916, 6597, 23866, 74400, 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-2*j+k,j)/(2*n-2*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,   1,   3,    6,   10,   15,   21, ...
  0,   3,   8,   16,   28,   45,   68, ...
  0,  10,  27,   54,   95,  155,  240, ...
  0,  46, 118,  228,  392,  631,  972, ...
  0, 244, 609, 1144, 1916, 3015, 4560, ...

Prog

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

Crossrefs

Columns k=0..2 give A000007, A143500, A384576.

Cf. A381566, A384581, A384582, A384583.

Sequence in context: A071569 A378321 A384865 * A392378 A261835 A384866

Adjacent sequences: A384577 A384578 A384579 * A384581 A384582 A384583

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 04 2025

Status

approved