A396992 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where column k is the expansion of B^k(x), where B(x) is the g.f. of A091713.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 15, 0, 1, 4, 15, 46, 99, 0, 1, 5, 24, 99, 330, 781, 0, 1, 6, 35, 180, 781, 2756, 7001, 0, 1, 7, 48, 295, 1564, 7001, 25782, 69253, 0, 1, 8, 63, 450, 2815, 15140, 69253, 263866, 742071, 0, 1, 9, 80, 651, 4694, 29465, 159580, 742071, 2909092, 8506775, 0

Offset

1,8

Links

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Formula

G.f. B(x) satisfies B(x) = x*(1 + B^l(x)), where B^l(x) denotes the l-th iterate of B.

Let a(n,k,l) = [x^n] B^k(x), where B^k(x) is the k-th iterate of B.

a(n,0,l) = 0^(n-1) and a(n,k,l) = a(n,k-1,l) + Sum_{j=1..n-1} a(j,k+l-1,l) * a(n-j,k-1,l) for k > 0.

A(n,k) = a(n,k,3).

Example

Square array begins:
  1,   1,    1,    1,     1,     1,     1, ...
  0,   1,    2,    3,     4,     5,     6, ...
  0,   3,    8,   15,    24,    35,    48, ...
  0,  15,   46,   99,   180,   295,   450, ...
  0,  99,  330,  781,  1564,  2815,  4694, ...
  0, 781, 2756, 7001, 15140, 29465, 53056, ...
  ...

Crossrefs

Column k=1 gives A091713.

Column k=3 gives A091713(n+1).

Cf. A128325, A396993, A396994, A396995.

Cf. A396971.

Sequence in context: A128888 A384801 A396412 * A384802 A380178 A384804

Adjacent sequences: A396987 A396988 A396989 * A396993 A396994 A396995

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 13 2026

Status

approved