A384977 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 A384974.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 8, 0, 1, 4, 12, 22, 25, 0, 1, 5, 18, 43, 75, 57, 0, 1, 6, 25, 72, 159, 212, 22, 0, 1, 7, 33, 110, 287, 516, 372, -1003, 0, 1, 8, 42, 158, 470, 1032, 1296, -1220, -9967, 0, 1, 9, 52, 217, 720, 1836, 3126, 378, -20271, -67627, 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(-3*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,     1, ...
  0,  1,   2,    3,    4,    5,     6, ...
  0,  3,   7,   12,   18,   25,    33, ...
  0,  8,  22,   43,   72,  110,   158, ...
  0, 25,  75,  159,  287,  470,   720, ...
  0, 57, 212,  516, 1032, 1836,  3018, ...
  0, 22, 372, 1296, 3126, 6295, 11353, ...

Prog

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-3*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, A384974.

Sequence in context: A384903 A255961 A297328 * A378289 A362079 A378292

Adjacent sequences: A384974 A384975 A384976 * A384978 A384979 A384980

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 14 2025

Status

approved