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A384903 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 A384898.

%I A384903 #16 Jun 12 2025 10:22:54
%S A384903 1,1,0,1,1,0,1,2,3,0,1,3,7,6,0,1,4,12,18,10,0,1,5,18,37,41,-39,0,1,6,
%T A384903 25,64,102,-22,-546,0,1,7,33,100,203,96,-1074,-3563,0,1,8,42,146,355,
%U A384903 372,-1419,-8332,-18918,0,1,9,52,203,570,876,-1338,-13974,-48606,-68472,0
%N A384903 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 A384898.
%F A384903 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-3*n+4*j+k-1,j-1) * b(n-j,3*j)/j. Then A(n,k) = b(n,-k).
%e A384903 Square array begins:
%e A384903   1,   1,   1,   1,   1,   1, ...
%e A384903   0,   1,   2,   3,   4,   5, ...
%e A384903   0,   3,   7,  12,  18,  25, ...
%e A384903   0,   6,  18,  37,  64, 100, ...
%e A384903   0,  10,  41, 102, 203, 355, ...
%e A384903   0, -39, -22,  96, 372, 876, ...
%o A384903 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-3*n+4*j+k-1,j-1)*b(n-j,3*j)/j));
%o A384903 a(n, k) = b(n, -k);
%Y A384903 Columns k=0..1 give A000007, A384898.
%Y A384903 Cf. A384901, A384902.
%Y A384903 Cf. A384865.
%K A384903 sign,tabl
%O A384903 0,8
%A A384903 _Seiichi Manyama_, Jun 12 2025