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A384626 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 A143426.

%I A384626 #10 Jun 05 2025 09:52:15
%S A384626 1,1,0,1,1,0,1,2,2,0,1,3,5,7,0,1,4,9,18,32,0,1,5,14,34,82,175,0,1,6,
%T A384626 20,56,156,442,1086,0,1,7,27,85,261,834,2699,7429,0,1,8,35,122,405,
%U A384626 1392,5027,18178,54994,0,1,9,44,168,597,2166,8310,33387,132664,435120,0
%N A384626 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 A143426.
%F A384626 A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(n-j+k,j)/(n-j+k) * A(n-j,2*j).
%e A384626 Square array begins:
%e A384626   1,    1,    1,    1,    1,     1,     1, ...
%e A384626   0,    1,    2,    3,    4,     5,     6, ...
%e A384626   0,    2,    5,    9,   14,    20,    27, ...
%e A384626   0,    7,   18,   34,   56,    85,   122, ...
%e A384626   0,   32,   82,  156,  261,   405,   597, ...
%e A384626   0,  175,  442,  834, 1392,  2166,  3216, ...
%e A384626   0, 1086, 2699, 5027, 8310, 12850, 19022, ...
%o A384626 (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 2*j)));
%Y A384626 Columns k=0..1 give A000007, A143426.
%K A384626 nonn,tabl
%O A384626 0,8
%A A384626 _Seiichi Manyama_, Jun 05 2025