A384654 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 A384650.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 10, 0, 1, 4, 9, 24, 69, 0, 1, 5, 14, 43, 162, 592, 0, 1, 6, 20, 68, 285, 1362, 6052, 0, 1, 7, 27, 100, 445, 2352, 13664, 70870, 0, 1, 8, 35, 140, 650, 3612, 23171, 157592, 928497, 0, 1, 9, 44, 189, 909, 5201, 34972, 263190, 2039543, 13404514, 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(5*n-4*j+k,j)/(5*n-4*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,    2,     5,     9,    14,    20,    27, ...
  0,   10,    24,    43,    68,   100,   140, ...
  0,   69,   162,   285,   445,   650,   909, ...
  0,  592,  1362,  2352,  3612,  5201,  7188, ...
  0, 6052, 13664, 23171, 34972 ,49540, 67433, ...

Prog

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

Crossrefs

Columns k=0..1 give A000007, A384650.

Cf. A379598, A384651, A384652, A384653.

Sequence in context: A091063 A384652 A384653 * A392379 A246935 A383042

Adjacent sequences: A384651 A384652 A384653 * A384655 A384656 A384657

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 06 2025

Status

approved