A384866 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 A213093.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 4, 0, 1, 4, 6, 10, 13, 0, 1, 5, 10, 19, 35, 62, 0, 1, 6, 15, 32, 69, 158, 297, 0, 1, 7, 21, 50, 119, 303, 760, 1523, 0, 1, 8, 28, 74, 190, 516, 1453, 3868, 8091, 0, 1, 9, 36, 105, 288, 821, 2462, 7359, 20487, 43243, 0

Offset

0,8

Comments

A(42,1) = -16825305705383790675462237694.

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(-4*n+5*j+k-1,j-1) * b(n-j,j)/j. Then A(n,k) = b(n,-k).

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,   4,  10,   19,   32,   50,   74, ...
  0,  13,  35,   69,  119,  190,  288, ...
  0,  62, 158,  303,  516,  821, 1248, ...
  0, 297, 760, 1453, 2462, 3900, 5913, ...

Prog

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-4*n+5*j+k-1, j-1)*b(n-j, j)/j));
a(n, k) = b(n, -k);

Crossrefs

Columns k=0..1 give A000007, A213093.

Sequence in context: A384580 A392378 A261835 * A384581 A384582 A384583

Adjacent sequences: A384863 A384864 A384865 * A384867 A384868 A384869

Keyword

tabl,sign

Author

Seiichi Manyama, Jun 11 2025

Status

approved