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A384945 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 A384942.

%I A384945 #9 Jun 13 2025 08:20:57
%S A384945 1,1,0,1,1,0,1,2,5,0,1,3,11,-5,0,1,4,18,0,-135,0,1,5,26,16,-255,-110,
%T A384945 0,1,6,35,44,-345,-540,3661,0,1,7,45,85,-389,-1230,5777,16440,0,1,8,
%U A384945 56,140,-370,-2100,5918,40452,-1375,0,1,9,68,210,-270,-3049,3784,67356,86065,-827075,0
%N A384945 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 A384942.
%F A384945 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,5*j)/j. Then A(n,k) = b(n,-k).
%e A384945 Square array begins:
%e A384945   1,    1,    1,     1,     1,     1,     1, ...
%e A384945   0,    1,    2,     3,     4,     5,     6, ...
%e A384945   0,    5,   11,    18,    26,    35,    45, ...
%e A384945   0,   -5,    0,    16,    44,    85,   140, ...
%e A384945   0, -135, -255,  -345,  -389,  -370,  -270, ...
%e A384945   0, -110, -540, -1230, -2100, -3049, -3954, ...
%e A384945   0, 3661, 5777,  5918,  3784,  -770, -7708, ...
%o A384945 (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, 5*j)/j));
%o A384945 a(n, k) = b(n, -k);
%Y A384945 Columns k=0..1 give A000007, A384942.
%Y A384945 Cf. A384899, A384901.
%K A384945 sign,tabl
%O A384945 0,8
%A A384945 _Seiichi Manyama_, Jun 13 2025