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A385018 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 A385014.

%I A385018 #10 Jun 15 2025 09:56:46
%S A385018 1,1,0,1,1,0,1,2,3,0,1,3,7,4,0,1,4,12,14,3,0,1,5,18,31,23,-15,0,1,6,
%T A385018 25,56,69,0,-118,0,1,7,33,90,151,84,-232,-336,0,1,8,42,134,280,288,
%U A385018 -234,-974,-595,0,1,9,52,189,468,676,54,-1749,-2681,1467,0,1,10,63,256,728,1326,895,-2232,-6585,-1306,20391,0
%N A385018 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 A385014.
%F A385018 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,2*j)/j. Then A(n,k) = b(n,-k).
%e A385018 Square array begins:
%e A385018   1,    1,    1,    1,   1,   1,    1, ...
%e A385018   0,    1,    2,    3,   4,   5,    6, ...
%e A385018   0,    3,    7,   12,  18,  25,   33, ...
%e A385018   0,    4,   14,   31,  56,  90,  134, ...
%e A385018   0,    3,   23,   69, 151, 280,  468, ...
%e A385018   0,  -15,    0,   84, 288, 676, 1326, ...
%e A385018   0, -118, -232, -234,  54, 895, 2653, ...
%o A385018 (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, 2*j)/j));
%o A385018 a(n, k) = b(n, -k);
%Y A385018 Columns k=0..1 give A000007, A385014.
%K A385018 sign,tabl
%O A385018 0,8
%A A385018 _Seiichi Manyama_, Jun 15 2025