A136552 a(n) = C(2*2^n + 2*n, n)*2^n/(2^n + n); a(n) = coefficient of x^n in Catalan(x)^(2*2^n).

1, 4, 44, 1120, 73112, 13931904, 8577576576, 18194461305856, 137735630840752320, 3788203438909701560320, 381994324029534476962777088, 141991478147899869433639040073728

Offset

0,2

Links

Table of n, a(n) for n=0..11.

Formula

G.f.: A(x) = Sum_{n>=0} 2^n * log( Catalan(2^n*x) )^n / n! where Catalan(x) = 2/(1+sqrt(1-4*x)).

Example

G.f.: A(x) = 1 + 4*x + 44*x^2 + 1120*x^3 + 73112*x^4 +...
This is a special application of the following identity.
Let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =
Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b).
Here F(x) = Catalan(x), q=2, m=2, b=0.

Prog

(PARI) {a(n)=binomial(2*2^n + 2*n, n)*2^n/(2^n + n)}
(PARI) {a(n)=polcoeff((2/(1+sqrt(1-4*x +x*O(x^n))))^(2*2^n), n)}
(PARI) {a(n)=polcoeff(sum(k=0, n, 2^k*log( 2/(1+sqrt(1-4*2^k*x+x*O(x^n))))^k/k!), n)}

Crossrefs

Cf. A136550, A136551.

Sequence in context: A348130 A255928 A137783 * A155556 A188456 A306372

Adjacent sequences: A136549 A136550 A136551 * A136553 A136554 A136555

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2008

Status

approved