A381594 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 A381601.
1, 1, 0, 1, 1, 0, 1, 2, 7, 0, 1, 3, 15, 79, 0, 1, 4, 24, 172, 1134, 0, 1, 5, 34, 280, 2475, 18953, 0, 1, 6, 45, 404, 4044, 41280, 353134, 0, 1, 7, 57, 545, 5863, 67365, 766291, 7154751, 0, 1, 8, 70, 704, 7955, 97620, 1246534, 15460284, 155181240, 0, 1, 9, 84, 882, 10344, 132486, 1801536, 25051422, 333896388, 3565276582, 0
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, ... 0, 7, 15, 24, 34, 45, 57, ... 0, 79, 172, 280, 404, 545, 704, ... 0, 1134, 2475, 4044, 5863, 7955, 10344, ... 0, 18953, 41280, 67365, 97620, 132486, 172434, ... 0, 353134, 766291, 1246534, 1801536, 2439615, 3169770, ...
Programs
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PARI
a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n+j+k, j)/(3*n+j+k)*a(n-j, 3*j)));
Formula
A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(3*n+j+k,j)/(3*n+j+k) * A(n-j,3*j).