A136578 - OEIS

%I #2 Mar 30 2012 18:37:08

%S 1,5,78,6527,3450452,12594729052,338284182093366,70004091118158663618,

%T 115159273597941035104859580,1536760523930850376685165570432060,

%U 168534058834325412618424268506407590697776

%N G.f.: A(x) = Sum_{n>=0} log( (1 + 2^n*x)*(1 + 3^n*x) )^n / n!.

%F a(n) = Sum_{k=0..n} C(2^k*3^(n-k), k) * C(2^k*3^(n-k), n-k).

%e G.f. A(x) = 1 + 5*x + 78*x^2 + 6527*x^3 + 3450452*x^4 +...

%e A(x) = 1 + log((1+2x)(1+3x)) + log((1+4x)(1+9x))^2/2! + log((1+8x)(1+27x))^3/3! +...

%e More generally: if Sum_{n>=0} (1+p^n*x)^b*(1+q^n*x)^d * log((1+p^n*x)*(1+q^n*x))^n /n! = Sum_{n>=0} a(n)*x^n then a(n) = Sum_{k=0..n} C(p^k*q^(n-k)+b, k) * C(p^k*q^(n-k)+d, n-k).

%o (PARI) {a(n)=polcoeff(sum(i=0,n,log((1+2^i*x)*(1+3^i*x)+x*O(x^n))^i/i!),n)}

%o (PARI) {a(n)=sum(k=0,n,binomial(2^k*3^(n-k),k)*binomial(2^k*3^(n-k),n-k))}

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 08 2008, Jan 23 2008