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A385019 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 A385015.

%I A385019 #11 Jun 15 2025 09:56:35
%S A385019 1,1,0,1,1,0,1,2,4,0,1,3,9,4,0,1,4,15,16,-13,0,1,5,22,37,-2,-81,0,1,6,
%T A385019 30,68,45,-156,-389,0,1,7,39,110,141,-165,-1028,-198,0,1,8,49,164,300,
%U A385019 -32,-1796,-1926,7455,0,1,9,60,231,537,336,-2460,-5499,10923,44515,0
%N A385019 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 A385015.
%F A385019 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,3*j)/j. Then A(n,k) = b(n,-k).
%e A385019 Square array begins:
%e A385019   1,    1,     1,     1,     1,     1,     1, ...
%e A385019   0,    1,     2,     3,     4,     5,     6, ...
%e A385019   0,    4,     9,    15,    22,    30,    39, ...
%e A385019   0,    4,    16,    37,    68,   110,   164, ...
%e A385019   0,  -13,    -2,    45,   141,   300,   537, ...
%e A385019   0,  -81,  -156,  -165,   -32,   336,  1050, ...
%e A385019   0, -389, -1028, -1796, -2460, -2655, -1863, ...
%o A385019 (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, 3*j)/j));
%o A385019 a(n, k) = b(n, -k);
%Y A385019 Columns k=0..1 give A000007, A385015.
%K A385019 sign,tabl
%O A385019 0,8
%A A385019 _Seiichi Manyama_, Jun 15 2025