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A384864 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 A213091.

%I A384864 #17 Jun 12 2025 08:29:45
%S A384864 1,1,0,1,1,0,1,2,1,0,1,3,3,2,0,1,4,6,6,4,0,1,5,10,13,13,11,0,1,6,15,
%T A384864 24,30,34,31,0,1,7,21,40,59,78,96,98,0,1,8,28,62,105,156,220,296,317,
%U A384864 0,1,9,36,91,174,286,442,669,952,1070,0,1,10,45,128,273,492,820,1336,2136,3182,3685,0
%N A384864 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 A213091.
%C A384864 A(80,1) = -862046378563961332360903978002479001059082.
%F A384864 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,j)/j. Then A(n,k) = b(n,-k).
%e A384864 Square array begins:
%e A384864   1,  1,  1,   1,   1,   1,    1, ...
%e A384864   0,  1,  2,   3,   4,   5,    6, ...
%e A384864   0,  1,  3,   6,  10,  15,   21, ...
%e A384864   0,  2,  6,  13,  24,  40,   62, ...
%e A384864   0,  4, 13,  30,  59, 105,  174, ...
%e A384864   0, 11, 34,  78, 156, 286,  492, ...
%e A384864   0, 31, 96, 220, 442, 820, 1437, ...
%o A384864 (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, j)/j));
%o A384864 a(n, k) = b(n, -k);
%Y A384864 Columns k=0..1 give A000007, A213091.
%K A384864 sign,tabl
%O A384864 0,8
%A A384864 _Seiichi Manyama_, Jun 11 2025