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.
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, 0, 1, 6, 35, 44, -345, -540, 3661, 0, 1, 7, 45, 85, -389, -1230, 5777, 16440, 0, 1, 8, 56, 140, -370, -2100, 5918, 40452, -1375, 0, 1, 9, 68, 210, -270, -3049, 3784, 67356, 86065, -827075, 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, -5, 0, 16, 44, 85, 140, ... 0, -135, -255, -345, -389, -370, -270, ... 0, -110, -540, -1230, -2100, -3049, -3954, ... 0, 3661, 5777, 5918, 3784, -770, -7708, ...
Programs
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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)); 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+2*j+k-1,j-1) * b(n-j,5*j)/j. Then A(n,k) = b(n,-k).