A177778 - OEIS

%I #2 Mar 30 2012 18:37:22

%S 1,2,12,160,4272,221648,22347648,4416360160,1724182065408,

%T 1336677590208512,2064038664552586752,6359502604300426739200,

%U 39136760890428640414851072,481344480930558145524346370048

%N E.g.f.: A(x) = Sum_{n>=0} 2^n/n!*Product_{k=0..n-1} L(2^k*x), where L(x) is the e.g.f. of A177777.

%e E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 160*x^3/3! + 4272*x^4/4! +...

%e Then e.g.f. A(x) is given by:

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

%e where L(x) is the e.g.f. of A177777:

%e . L(x) = x + 2*x^2/2! + 12*x^3/3! + 152*x^4/4! + 3640*x^5/5! +...

%e . L(x) = x*d/dx log( Sum_{n>=0} 2^(n(n-1)/2)*x^n/n! )

%e and satisfies:

%e . L(x)/x = 1 + L(x) + L(x)L(2x)/2! + L(x)L(2x)L(4x)/3! + L(x)L(2x)L(4x)L(8x)/4! +...

%o (PARI) {a(n,q=2)=local(Lq=x+x^2,A); for(i=1,n,Lq=x*sum(m=0,n,(q-1)^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n))))); A=sum(m=0,n,2^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n))));n!*polcoeff(A,n)}

%Y Cf. A177777, A177780.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 20 2010