A394047 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 A393889.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 27, 0, 1, 4, 12, 60, 387, 0, 1, 5, 18, 100, 837, 7317, 0, 1, 6, 25, 148, 1359, 15570, 168237, 0, 1, 7, 33, 205, 1963, 24867, 354159, 4503789, 0, 1, 8, 42, 272, 2660, 35328, 559440, 9408852, 136727055, 0, 1, 9, 52, 350, 3462, 47086, 785916, 14747022, 284015997, 4625789067, 0

Offset

0,8

Links

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

Formula

A(n,0) = 0^n.

A(0,1) = A(1,1) = 1; A(n,1) = 3 * Sum_{j=1..n-1} A(j,1) * A(n-j,j).

For k > 1, A(0,k) = 1; A(n,k) = (1/n) * Sum_{j=1..n} ((k+1)*j-n) * A(j,1) * A(n-j,k).

Example

Square array begins:
  1,    1,     1,     1,     1,     1, ...
  0,    1,     2,     3,     4,     5, ...
  0,    3,     7,    12,    18,    25, ...
  0,   27,    60,   100,   148,   205, ...
  0,  387,   837,  1359,  1963,  2660, ...
  0, 7317, 15570, 24867, 35328, 47086, ...

Prog

(PARI) a(n, k, m=3) = if(n*k<=1, k^n, if(k==1, m*sum(j=1, n-1, a(j, 1)*a(n-j, j)), sum(j=1, n, ((k+1)*j-n)*a(j, 1)*a(n-j, k))/n));

Crossrefs

Columns k=0..1 give A000007, A393889.

Cf. A392378, A392379, A394901.

Sequence in context: A379599 A384681 A384777 * A055137 A143325 A307910

Adjacent sequences: A394044 A394045 A394046 * A394048 A394049 A394050

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 06 2026

Status

approved