A392379 - OEIS

%I #16 Apr 06 2026 09:35:38

%S 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,

%T 6,20,76,438,3256,23320,0,1,7,27,110,657,5352,50224,427168,0,1,8,35,

%U 152,925,7832,81212,909712,8903096,0,1,9,44,203,1251,10761,116856,1454064,18802976,207404528,0

%N 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.

%F A(n,0) = 0^n.

%F A(0,1) = A(1,1) = 1;   A(n,1) =     2 * Sum_{j=1..n-1} A(j,1) * A(n-j,j).

%F 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).

%e Square array begins:

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

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

%e   0,    2,    5,    9,   14,    20, ...

%e   0,   12,   28,   49,   76,   110, ...

%e   0,  116,  260,  438,  657,   925, ...

%e   0, 1488, 3256, 5352, 7832, 10761, ...

%o (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));

%Y Columns k=0..1 give A000007, A393888.

%Y Cf. A392378, A394047, A394901.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Apr 06 2026