A136554 - OEIS

%I #4 Jul 03 2012 21:06:25

%S 1,3,10,82,2304,232088,81639942,99425060368,421915147527984,

%T 6313762292901492960,337457827116687464134048,

%U 65175276571204939272971781496,45944813538624773942727094008288680

%N G.f.: A(x) = Sum_{n>=0} log( (1 + x)*(1 + 2^n*x) )^n / n!.

%F a(n) = Sum_{k=0..n} C(2^k, k)*C(2^k, n-k).

%F G.f.: A(x) = Sum_{n>=0} C(2^n,n) * x^n * (1+x)^(2^n).

%e G.f.: A(x) = 1 + 3*x + 10*x^2 + 82*x^3 + 2304*x^4 + 232088*x^5 +...;

%e A(x) = 1 + log((1+x)*(1+2*x)) + log((1+x)*(1+4*x))^2/2! + log((1+x)*(1+8*x))^3/3! + log((1+x)*(1+16*x))^4/4! +...

%e Surprisingly, this sum yields a series in x with only integer coefficients.

%o (PARI) {a(n)=polcoeff(sum(i=0,n,log((1+x)*(1+2^i*x)+x*O(x^n))^i/i!),n)}

%o (PARI) {a(n)=sum(k=0,n,binomial(2^k,k)*binomial(2^k,n-k))}

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 06 2008, Jan 07 2008