A088926 Main diagonal of table A088925, which lists coefficients T(n,k) of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/(1-x-y) + xy*f(x,y)^3.

1, 3, 21, 212, 2617, 36345, 544080, 8577378, 140456625, 2368062095, 40859183247, 718386164556, 12829418522056, 232153200359592, 4248457201595622, 78508329463480160, 1463164022514939392, 27474112707608092672

Offset: 0

Author

Paul D. Hanna, Oct 23 2003

Keywords

• nonn

Comments

The g.f. for A001764 satisfies: g(x) = 1 + x*g(x)^3.

Crossrefs

Cf. A088925 (table), A088927 (antidiagonal sums), A001764.

Programs

Mathematica

Table[Sum[Binomial[2*n, 2*i] * Binomial[2*n - 2*i, n - i]*(3*i)!/(i!*(2*i + 1)!), {i, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 10 2020 *)

Formula

a(n) = sum(i=0, n, C(2n, 2i)*C(2n-2i, n-i)*A001764(i) ), where A001764(i)=(3i)!/[i!(2i+1)! ] (from Michael Somos).

a(n) ~ (4 + 3*sqrt(3))^(2*n + 2) / (Pi * 3^(7/4) * n^2 * 2^(2*n + 4)). - Vaclav Kotesovec, Oct 10 2020