A361840 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 - x)^k)^(1/3).
1, 1, 3, 1, 3, 18, 1, 3, 15, 126, 1, 3, 12, 90, 945, 1, 3, 9, 57, 585, 7371, 1, 3, 6, 27, 297, 3969, 58968, 1, 3, 3, 0, 78, 1629, 27657, 480168, 1, 3, 0, -24, -75, 207, 9216, 196290, 3961386, 1, 3, -3, -45, -165, -438, 459, 53217, 1411965, 33011550
Offset: 0
Examples
Square array begins:
1, 1, 1, 1, 1, 1, ...
3, 3, 3, 3, 3, 3, ...
18, 15, 12, 9, 6, 3, ...
126, 90, 57, 27, 0, -24, ...
945, 585, 297, 78, -75, -165, ...
7371, 3969, 1629, 207, -438, -444, ...
Crossrefs
Programs
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PARI
T(n, k) = (-1)^n*sum(j=0, n, 9^j*binomial(-1/3, j)*binomial(k*j, n-j));
Formula
n*T(n,k) = 3 * Sum_{j=0..k} (-1)^j * binomial(k,j)*(3*n-2-2*j)*T(n-1-j,k) for n > k.
T(n,k) = (-1)^n * Sum_{j=0..n} 9^j * binomial(-1/3,j) * binomial(k*j,n-j).