A396993 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 A196523.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 10, 28, 0, 1, 4, 18, 77, 262, 0, 1, 5, 28, 153, 760, 2944, 0, 1, 6, 40, 262, 1608, 8846, 37666, 0, 1, 7, 54, 410, 2944, 19566, 116140, 532276, 0, 1, 8, 70, 603, 4930, 37666, 265722, 1675034, 8151322, 0, 1, 9, 88, 847, 7752, 66530, 532276, 3938097, 26089876, 133562194, 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,4).

Example

Square array begins:
  1,    1,    1,     1,     1,     1,      1, ...
  0,    1,    2,     3,     4,     5,      6, ...
  0,    4,   10,    18,    28,    40,     54, ...
  0,   28,   77,   153,   262,   410,    603, ...
  0,  262,  760,  1608,  2944,  4930,   7752, ...
  0, 2944, 8846, 19566, 37666, 66530, 110484, ...
  ...

Crossrefs

Column k=1 gives A196523.

Column k=4 gives A196523(n+1).

Cf. A128325, A396992, A396994, A396995.

Cf. A396972.

Sequence in context: A394901 A309148 A396413 * A396447 A397000 A396972

Adjacent sequences: A396989 A396990 A396992 * A396994 A396995 A396996

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 13 2026

Status

approved