A378320 - OEIS

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A378320

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(2*n-2*r+k,r) * binomial(r,n-r)/(2*n-2*r+k) for k > 0.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 3, 0, 1, 5, 10, 13, 11, 6, 0, 1, 6, 15, 24, 27, 22, 11, 0, 1, 7, 21, 40, 55, 57, 44, 22, 0, 1, 8, 28, 62, 100, 124, 121, 90, 44, 0, 1, 9, 36, 91, 168, 241, 278, 258, 187, 90, 0, 1, 10, 45, 128, 266, 432, 570, 620, 555, 392, 187, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..77.

FORMULA

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x^2 * A_k(x)^(2/k) )^k for k > 0.

G.f. of column k: B(x)^k where B(x) is the g.f. of A007477.

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

EXAMPLE

Square array begins:

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