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A384946 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 A384943.

%I A384946 #9 Jun 13 2025 08:20:35
%S A384946 1,1,0,1,1,0,1,2,6,0,1,3,13,-9,0,1,4,21,-6,-244,0,1,5,30,10,-470,-39,
%T A384946 0,1,6,40,40,-660,-674,11262,0,1,7,51,85,-795,-1824,19599,36971,0,1,8,
%U A384946 63,146,-855,-3384,24171,100390,-268890,0,1,9,76,224,-819,-5224,24318,180627,-268456,-3724293,0
%N A384946 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 A384943.
%F A384946 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-n+2*j+k-1,j-1) * b(n-j,6*j)/j. Then A(n,k) = b(n,-k).
%e A384946 Square array begins:
%e A384946   1,     1,     1,     1,     1,     1,     1, ...
%e A384946   0,     1,     2,     3,     4,     5,     6, ...
%e A384946   0,     6,    13,    21,    30,    40,    51, ...
%e A384946   0,    -9,    -6,    10,    40,    85,   146, ...
%e A384946   0,  -244,  -470,  -660,  -795,  -855,  -819, ...
%e A384946   0,   -39,  -674, -1824, -3384, -5224, -7188, ...
%e A384946   0, 11262, 19599, 24171, 24318, 19590,  9778, ...
%o A384946 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+2*j+k-1, j-1)*b(n-j, 6*j)/j));
%o A384946 a(n, k) = b(n, -k);
%Y A384946 Columns k=0..1 give A000007, A384943.
%Y A384946 Cf. A384899, A384901.
%K A384946 sign,tabl
%O A384946 0,8
%A A384946 _Seiichi Manyama_, Jun 13 2025