A392379 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 A393888.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 12, 0, 1, 4, 9, 28, 116, 0, 1, 5, 14, 49, 260, 1488, 0, 1, 6, 20, 76, 438, 3256, 23320, 0, 1, 7, 27, 110, 657, 5352, 50224, 427168, 0, 1, 8, 35, 152, 925, 7832, 81212, 909712, 8903096, 0, 1, 9, 44, 203, 1251, 10761, 116856, 1454064, 18802976, 207404528, 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) = 2 * 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,    2,    5,    9,   14,    20, ...
  0,   12,   28,   49,   76,   110, ...
  0,  116,  260,  438,  657,   925, ...
  0, 1488, 3256, 5352, 7832, 10761, ...

Prog

(PARI) a(n, k, m=2) = 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, A393888.

Cf. A392378, A394047, A394901.

Sequence in context: A384652 A384653 A384654 * A246935 A383042 A342120

Adjacent sequences: A392376 A392377 A392378 * A392380 A392381 A392382

Keyword

nonn,tabl

Author

Seiichi Manyama, Apr 06 2026

Status

approved