A384681 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 A384680.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 15, 0, 1, 4, 12, 36, 100, 0, 1, 5, 18, 64, 239, 805, 0, 1, 6, 25, 100, 426, 1900, 7442, 0, 1, 7, 33, 145, 671, 3357, 17319, 76750, 0, 1, 8, 42, 200, 985, 5260, 30228, 176214, 866818, 0, 1, 9, 52, 266, 1380, 7706, 46880, 303687, 1965938, 10586499, 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(3*n-j+k,j)/(3*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,    3,     7,    12,    18,    25,    33, ...
  0,   15,    36,    64,   100,   145,   200, ...
  0,  100,   239,   426,   671,   985,  1380, ...
  0,  805,  1900,  3357,  5260,  7706, 10806, ...
  0, 7442, 17319, 30228, 46880, 68115, 94918, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A384680.

Cf. A384581, A384652.

Sequence in context: A362079 A378292 A379599 * A384777 A394047 A055137

Adjacent sequences: A384678 A384679 A384680 * A384682 A384683 A384684

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 06 2025

Status

approved