A136552 - OEIS

%I #4 Jun 13 2015 09:41:26

%S 1,4,44,1120,73112,13931904,8577576576,18194461305856,

%T 137735630840752320,3788203438909701560320,

%U 381994324029534476962777088,141991478147899869433639040073728

%N 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).

%F 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)).

%e G.f.: A(x) = 1 + 4*x + 44*x^2 + 1120*x^3 + 73112*x^4 +...

%e This is a special application of the following identity.

%e Let F(x) be a formal power series in x such that F(0)=1, then

%e Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =

%e Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b).

%e Here F(x) = Catalan(x), q=2, m=2, b=0.

%o (PARI) {a(n)=binomial(2*2^n + 2*n, n)*2^n/(2^n + n)}

%o (PARI) {a(n)=polcoeff((2/(1+sqrt(1-4*x +x*O(x^n))))^(2*2^n),n)}

%o (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)}

%Y Cf. A136550, A136551.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 05 2008