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A384976 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 A384951.

%I A384976 #9 Jun 14 2025 10:37:45
%S A384976 1,1,0,1,1,0,1,2,2,0,1,3,5,3,0,1,4,9,10,5,0,1,5,14,22,20,6,0,1,6,20,
%T A384976 40,51,34,2,0,1,7,27,65,105,105,45,-20,0,1,8,35,98,190,248,188,18,
%U A384976 -102,0,1,9,44,140,315,501,526,255,-175,-312,0,1,10,54,192,490,912,1200,956,63,-836,-795,0
%N A384976 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 A384951.
%F A384976 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-n+j+k-1,j-1) * b(n-j,j)/j. Then A(n,k) = b(n,-k).
%e A384976 Square array begins:
%e A384976   1, 1,  1,   1,   1,    1,    1, ...
%e A384976   0, 1,  2,   3,   4,    5,    6, ...
%e A384976   0, 2,  5,   9,  14,   20,   27, ...
%e A384976   0, 3, 10,  22,  40,   65,   98, ...
%e A384976   0, 5, 20,  51, 105,  190,  315, ...
%e A384976   0, 6, 34, 105, 248,  501,  912, ...
%e A384976   0, 2, 45, 188, 526, 1200, 2408, ...
%o A384976 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+j+k-1, j-1)*b(n-j, j)/j));
%o A384976 a(n, k) = b(n, -k);
%Y A384976 Columns k=0..1 give A000007, A384951.
%K A384976 sign,tabl
%O A384976 0,8
%A A384976 _Seiichi Manyama_, Jun 14 2025