A396993 - OEIS

%I #16 Jun 13 2026 09:39:29

%S 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,

%T 0,1,6,40,262,1608,8846,37666,0,1,7,54,410,2944,19566,116140,532276,0,

%U 1,8,70,603,4930,37666,265722,1675034,8151322,0,1,9,88,847,7752,66530,532276,3938097,26089876,133562194,0

%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 A196523.

%H Seiichi Manyama, <a href="/A396993/b396993.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,4).

%e Square array begins:

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

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

%e   0,    4,   10,    18,    28,    40,     54, ...

%e   0,   28,   77,   153,   262,   410,    603, ...

%e   0,  262,  760,  1608,  2944,  4930,   7752, ...

%e   0, 2944, 8846, 19566, 37666, 66530, 110484, ...

%e   ...

%Y Column k=1 gives A196523.

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

%Y Cf. A128325, A396992, A396994, A396995.

%Y Cf. A396972.

%K nonn,tabl

%O 1,8

%A _Seiichi Manyama_, Jun 13 2026