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A384902 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 A384897.

%I A384902 #13 Jun 12 2025 10:23:00
%S A384902 1,1,0,1,1,0,1,2,3,0,1,3,7,3,0,1,4,12,12,-8,0,1,5,18,28,-1,-81,0,1,6,
%T A384902 25,52,30,-160,-462,0,1,7,33,85,95,-201,-1125,-1140,0,1,8,42,128,205,
%U A384902 -156,-1932,-3738,1662,0,1,9,52,182,372,36,-2760,-8073,-2150,42210,0
%N A384902 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 A384897.
%F A384902 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-2*n+3*j+k-1,j-1) * b(n-j,3*j)/j. Then A(n,k) = b(n,-k).
%e A384902 Square array begins:
%e A384902   1,   1,    1,    1,    1,   1, ...
%e A384902   0,   1,    2,    3,    4,   5, ...
%e A384902   0,   3,    7,   12,   18,  25, ...
%e A384902   0,   3,   12,   28,   52,  85, ...
%e A384902   0,  -8,   -1,   30,   95, 205, ...
%e A384902   0, -81, -160, -201, -156,  36, ...
%o A384902 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-2*n+3*j+k-1,j-1)*b(n-j,3*j)/j));
%o A384902 a(n, k) = b(n, -k);
%Y A384902 Columns k=0..1 give A000007, A384897.
%K A384902 sign,tabl
%O A384902 0,8
%A A384902 _Seiichi Manyama_, Jun 12 2025