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%I #9 Mar 30 2012 18:37:33
%S 1,2,4,16,144,768,10560,92160,1585920,21127680,419973120,7431782400,
%T 177081569280,3923981107200,105929096232960,2868863206809600,
%U 87449689674547200,2742391916199936000,94359281224797388800,3356687705428721664000
%N E.g.f.: A(x) = Sum_{n>=0} 2^n * binomial(x^n + n-1, n).
%C The definition of the e.g.f. A(x) is an application of the identity Sum_{n>=0} (-1)^n*log(1 - q^n*x)^n*y^n/n! = Sum_{n>=0} binomial(q^n*y +n-1, n)*x^n at x=2, y=1, q=x.
%F E.g.f.: A(x) = Sum_{n>=0} (-1)^n * log(1 - 2*x^n)^n/n!.
%F E.g.f.: A(x) = Sum_{n>=0} 2^n * Sum_{k=0..n} |Stirling1(n,k)|*x^(n*k)/n!.
%e E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 16*x^3/3! + 144*x^4/4! + 768*x^5/5! +...
%e Series expansions:
%e A(x) = 1 - log(1 - 2*x) + log(1 - 2*x^2)^2/2! - log(1 - 2*x^3)^3/3! + log(1 - 2*x^4)^4/4! +...+ (-1)^n*log(1 - 2*x^n)^n/n! +...
%e A(x) = 1 + 2*x + 2^2*x^2*(x^2+1)/2! + 2^3*x^3*(x^3+1)*(x^3+2)/3! + 2^4*x^4*(x^4+1)*(x^4+2)*(x^4+3)/4! +...+ 2^n*binomial(x^n + n-1, n) +...
%o (PARI) {a(n)=local(A=1+x); A=sum(m=0, n, 2^n*binomial(x^m+n-1 +x*O(x^n), m)); n!*polcoeff(A, n)}
%o (PARI) {a(n)=local(A=1+x); A=1+sum(m=1, n, (-1)^m*log(1-2*x^m +x*O(x^n))^m/m!); n!*polcoeff(A, n)}
%o (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, 2^m*sum(k=0, m, abs(Stirling1(m, k))*x^(m*k))/m!)); n!*polcoeff(A, n)}
%Y Cf. A191461.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 17 2011
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