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%I #3 Nov 12 2012 17:48:58
%S 1,1,4,24,296,4280,79032,1978368,57803776,1949421888,78945675200,
%T 3678493774560,190462000632576,11112878148649472,730288018660087552,
%U 52727783838765181440,4148774572438335014912,358041540338404096024576,33700760914469383117799424
%N E.g.f.: Sum_{n>=0} log(1 + x*exp(2*n*x))^n / n!.
%F E.g.f. satisfies the identities:
%F (1) Sum_{n>=0} binomial(exp(2*n*x),n) * x^n.
%F (2) Sum_{n>=0} [Product_{k=0..n-1} (exp(2*n*x) - k)] * x^n/n!.
%F (3) Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * exp(2*n*k*x) / n!.
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 24*x^3/3! + 296*x^4/4! + 4280*x^5/5! +...
%e where the e.g.f. equals the series:
%e (0) A(x) = 1 + log(1+x*exp(2*x)) + log(1+x*exp(4*x))^2/2! + log(1+x*exp(6*x))^3/3! + log(1+x*exp(8*x))^4/4! + log(1+x*exp(10*x))^5/5! +...
%e (1) A(x) = 1 + binomial(exp(2*x),1)*x + binomial(exp(4*x),2)*x^2 + binomial(exp(6*x),3)*x^3 + binomial(exp(8*x),4)*x^4 + binomial(exp(10*x),5)*x^5 +...
%e (2) A(x) = 1 + exp(2*x)*x + exp(4*x)*(exp(4*x)-1)*x^2/2! + exp(6*x)*(exp(6*x)-1)*(exp(6*x)-2)*x^3/3! + exp(8*x)*(exp(8*x)-1)*(exp(8*x)-2)*(exp(8*x)-3)*x^4/4! +...
%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,log(1+x*exp(2*m*x+x*O(x^n)))^m/m!),n)}
%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,binomial(exp(2*m*x+x*O(x^n)),m)*x^m),n)}
%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n, prod(k=0,m-1, (exp(2*m*x +x*O(x^n)) - k)) * x^m/m!),n)}
%o (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o {a(n)=local(A=1+x); A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*exp(2*m*k*x+x*O(x^n)))*x^m/m!); n!*polcoeff(A, n)}
%o for(n=0, 31, print1(a(n), ", "))
%Y Cf. A216839.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 12 2012
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