A378318 - OEIS

A378318

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) * binomial(3*r+k,n)/(3*r+k) for k > 0.

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 30, 0, 1, 8, 30, 84, 170, 0, 1, 10, 48, 170, 496, 1050, 0, 1, 12, 70, 296, 1050, 3140, 6846, 0, 1, 14, 96, 470, 1920, 6846, 20832, 46374, 0, 1, 16, 126, 700, 3210, 12936, 46374, 142932, 323154, 0, 1, 18, 160, 994, 5040, 22402, 89712, 323154, 1005856, 2301618, 0

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OFFSET

0,5

LINKS

Table of n, a(n) for n=0..65.

FORMULA

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x * A_k(x)^(3/k) )^k for k > 0.

G.f. of column k: B(x)^k where B(x) is the g.f. of A366266.

B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k+2) for n > 0.

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, 1, ...