A384580 - OEIS

A384580

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 A143500.

%I #12 Jun 04 2025 09:52:51

%S 1,1,0,1,1,0,1,2,1,0,1,3,3,3,0,1,4,6,8,10,0,1,5,10,16,27,46,0,1,6,15,

%T 28,54,118,244,0,1,7,21,45,95,228,609,1481,0,1,8,28,68,155,392,1144,

%U 3602,10020,0,1,9,36,98,240,631,1916,6597,23866,74400,0

%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 A143500.

%F A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+k,j)/(2*n-2*j+k) * A(n-j,j).

%e Square array begins:

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

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

%e 0, 1, 3, 6, 10, 15, 21, ...

%e 0, 3, 8, 16, 28, 45, 68, ...

%e 0, 10, 27, 54, 95, 155, 240, ...

%e 0, 46, 118, 228, 392, 631, 972, ...

%e 0, 244, 609, 1144, 1916, 3015, 4560, ...

%o (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, j)));

%Y Columns k=0..2 give A000007, A143500, A384576.

%Y Cf. A381566, A384581, A384582, A384583.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Jun 04 2025