A396992 - OEIS

%I #19 Jun 13 2026 09:39:40

%S 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,

%T 6,35,180,781,2756,7001,0,1,7,48,295,1564,7001,25782,69253,0,1,8,63,

%U 450,2815,15140,69253,263866,742071,0,1,9,80,651,4694,29465,159580,742071,2909092,8506775,0

%N 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.

%H Seiichi Manyama, <a href="/A396992/b396992.txt">Antidiagonals n = 1..140, flattened</a>

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

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

%F 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.

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

%e Square array begins:

%e   1,   1,    1,    1,     1,     1,     1, ...

%e   0,   1,    2,    3,     4,     5,     6, ...

%e   0,   3,    8,   15,    24,    35,    48, ...

%e   0,  15,   46,   99,   180,   295,   450, ...

%e   0,  99,  330,  781,  1564,  2815,  4694, ...

%e   0, 781, 2756, 7001, 15140, 29465, 53056, ...

%e   ...

%Y Column k=1 gives A091713.

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

%Y Cf. A128325, A396993, A396994, A396995.

%Y Cf. A396971.

%K nonn,tabl

%O 1,8

%A _Seiichi Manyama_, Jun 13 2026