A384900 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 A384895.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 4, 0, 1, 5, 14, 22, 18, -3, 0, 1, 6, 20, 40, 48, 14, -50, 0, 1, 7, 27, 65, 101, 72, -81, -237, 0, 1, 8, 35, 98, 185, 200, -37, -562, -872, 0, 1, 9, 44, 140, 309, 436, 174, -873, -2420, -2375, 0

Offset

0,8

Links

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

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-2*n+3*j+k-1,j-1) * b(n-j,2*j)/j. Then A(n,k) = b(n,-k).

Example

Square array begins:
  1,  1,  1,  1,   1,   1, ...
  0,  1,  2,  3,   4,   5, ...
  0,  2,  5,  9,  14,  20, ...
  0,  3, 10, 22,  40,  65, ...
  0,  4, 18, 48, 101, 185, ...
  0, -3, 14, 72, 200, 436, ...

Prog

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-2*n+3*j+k-1, j-1)*b(n-j, 2*j)/j));
a(n, k) = b(n, -k);

Crossrefs

Columns k=0..1 give A000007, A384895.

Sequence in context: A182888 A317205 A384899 * A296068 A384976 A144064

Adjacent sequences: A384897 A384898 A384899 * A384901 A384902 A384903

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 12 2025

Status

approved