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%I #8 Feb 11 2013 00:30:02
%S 1,2,14,153,2262,42120,945823,24870937,749702348,25490350284,
%T 965219424913,40287663094503,1837912330721162,90988147658574582,
%U 4858595700832370019,278371180944699911227,17034913752075240500920,1108950617553341656112312,76523438074756638449399565
%N E.g.f.: Sum_{n>=0} log( Sum_{k>=0} (n+k)^k*x^k/k! )^n / n!.
%C Compare to the trivial identities:
%C (1) Sum_{n>=0} log( Sum_{k>=0} n^k*x^k/k! )^n/n! = Sum_{n>=0} n^n*x^n/n!;
%C (2) Sum_{n>=0} log( Sum_{k>=0} k^k*x^k/k! )^n/n! = Sum_{n>=0} n^n*x^n/n!.
%F E.g.f.: Sum_{n>=0} log( (LambertW(-x)/(-x))^n / (1+LambertW(-x)) )^n / n!.
%F E.g.f.: Sum_{n>=0} [ Sum_{k>=1} (k^(k-1)*n + A001865(k))*x^k/k! ]^n / n!.
%e E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 153*x^3/3! + 2262*x^4/4! + 42120*x^5/5! +...
%e such that
%e A(x) = 1 + log(F(x,1)) + log(F(x,2))^2/2! + log(F(x,3))^3/3! + log(F(x,4))^4/4! +...
%e where
%e F(x,n) = 1 + (n+1)*x + (n+2)^2*x^2/2! + (n+3)^3*x^3/3! + (n+4)^4*x^4/4! + (n+5)^5*x^5/5! +...+ (n+k)^k*x^k/k! +...
%e Also,
%e A(x) = 1 + G(x,1) + G(x,2)^2/2! + G(x,3)^3/3! + G(x,4)^4/4! +...+ G(x,n)^n/n! +...
%e where G(x,n) = log( (LambertW(-x)/(-x))^n / (1+LambertW(-x)) ):
%e G(x,n) = (n+1)*x + (2*n+3)*x^2/2! + (9*n+17)*x^3/3! + (64*n+142)*x^4/4! + (625*n+1569)*x^5/5! +...+ (k^(k-1)*n + A001865(k))*x^k/k! +...
%e Related expansion:
%e Sum_{n>=0} n^n*log(LambertW(-x)/(-x))^n/n! = 1/(1+LambertW(LambertW(-x)));
%e 1/(1+LambertW(LambertW(-x))) = 1 + x + 6*x^2/2! + 60*x^3/3! + 836*x^4/4! +...
%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,log(sum(k=0,n,(m+k)^k*x^k/k! +x*O(x^n)))^m/m!),n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A000169, A001865.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 10 2013
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