A378289 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n+r+k,r) * binomial(r,n-r)/(n+r+k) for k > 0.
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 10, 0, 1, 4, 12, 26, 38, 0, 1, 5, 18, 49, 105, 154, 0, 1, 6, 25, 80, 210, 444, 654, 0, 1, 7, 33, 120, 363, 927, 1944, 2871, 0, 1, 8, 42, 170, 575, 1672, 4191, 8734, 12925, 0, 1, 9, 52, 231, 858, 2761, 7810, 19305, 40040, 59345, 0
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, ... 0, 3, 7, 12, 18, 25, 33, ... 0, 10, 26, 49, 80, 120, 170, ... 0, 38, 105, 210, 363, 575, 858, ... 0, 154, 444, 927, 1672, 2761, 4290, ... 0, 654, 1944, 4191, 7810, 13325, 21385, ...
Crossrefs
Programs
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PARI
T(n, k, t=2, u=1) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k))); matrix(7, 7, n, k, T(n-1, k-1))
Formula
G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + x * A_k(x)^(1/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A001002.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-2,k+2) for n > 1.