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A384977 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 A384974.

%I A384977 #9 Jun 14 2025 10:37:30
%S A384977 1,1,0,1,1,0,1,2,3,0,1,3,7,8,0,1,4,12,22,25,0,1,5,18,43,75,57,0,1,6,
%T A384977 25,72,159,212,22,0,1,7,33,110,287,516,372,-1003,0,1,8,42,158,470,
%U A384977 1032,1296,-1220,-9967,0,1,9,52,217,720,1836,3126,378,-20271,-67627,0
%N A384977 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 A384974.
%F A384977 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-3*n+3*j+k-1,j-1) * b(n-j,2*j)/j. Then A(n,k) = b(n,-k).
%e A384977 Square array begins:
%e A384977   1,  1,   1,    1,    1,    1,     1, ...
%e A384977   0,  1,   2,    3,    4,    5,     6, ...
%e A384977   0,  3,   7,   12,   18,   25,    33, ...
%e A384977   0,  8,  22,   43,   72,  110,   158, ...
%e A384977   0, 25,  75,  159,  287,  470,   720, ...
%e A384977   0, 57, 212,  516, 1032, 1836,  3018, ...
%e A384977   0, 22, 372, 1296, 3126, 6295, 11353, ...
%o A384977 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-3*n+3*j+k-1, j-1)*b(n-j, 2*j)/j));
%o A384977 a(n, k) = b(n, -k);
%Y A384977 Columns k=0..1 give A000007, A384974.
%K A384977 sign,tabl
%O A384977 0,8
%A A384977 _Seiichi Manyama_, Jun 14 2025