A385020 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 A385016.
1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, 3, 0, 1, 4, 18, 16, -51, 0, 1, 5, 26, 40, -71, -190, 0, 1, 6, 35, 76, -45, -452, -401, 0, 1, 7, 45, 125, 43, -702, -1683, 3672, 0, 1, 8, 56, 188, 210, -836, -3784, 4336, 51925, 0, 1, 9, 68, 266, 474, -729, -6480, 144, 108645, 151539, 0
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, ... 0, 5, 11, 18, 26, 35, 45, ... 0, 3, 16, 40, 76, 125, 188, ... 0, -51, -71, -45, 43, 210, 474, ... 0, -190, -452, -702, -836, -729, -234, ... 0, -401, -1683, -3784, -6480, -9360, -11800, ...
Programs
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PARI
b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+j+k-1, j-1)*b(n-j, 4*j)/j)); a(n, k) = b(n, -k);
Formula
Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-n+j+k-1,j-1) * b(n-j,4*j)/j. Then A(n,k) = b(n,-k).