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A216839 E.g.f.: Sum_{n>=0} log(1 + x*exp(n*x))^n / n!. 3

%I

%S 1,1,2,9,64,515,6126,87332,1408352,28357821,656029450,16616305354,

%T 486491747952,16101080888763,572203757798414,22348109637703800,

%U 973262507935361632,45353465796372720729,2238286744709428606866,120361307277708751011502

%N E.g.f.: Sum_{n>=0} log(1 + x*exp(n*x))^n / n!.

%C Note that a(32)-a(42), a(57)-a(69), ... are negative, see b-file. - _Vaclav Kotesovec_, Nov 05 2014

%H Vaclav Kotesovec, <a href="/A216839/b216839.txt">Table of n, a(n) for n = 0..250</a>

%F E.g.f.: Sum_{n>=0} binomial(exp(n*x),n) * x^n.

%F E.g.f.: Sum_{n>=0} [Product_{k=0..n-1} (exp(n*x) - k)] * x^n/n!.

%F E.g.f.: Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * exp(n*k*x) / n!.

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 515*x^5/5! +...

%e where the g.f. satisfies the identities:

%e A(x) = 1 + log(1+x*exp(x)) + log(1+x*exp(2*x))^2/2! + log(1+x*exp(3*x))^3/3! + log(1+x*exp(4*x))^4/4! + log(1+x*exp(5*x))^5/5! +...

%e A(x) = 1 + binomial(exp(x),1)*x + binomial(exp(2*x),2)*x^2 + binomial(exp(3*x),3)*x^3 + binomial(exp(4*x),4)*x^4 + binomial(exp(5*x),5)*x^5 +...

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

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

%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,binomial(exp(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(m*x +x*O(x^n)) - k)) * x^m/m!),n)}

%o for(n=0,31,print1(a(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(m*k*x+x*O(x^n)))*x^m/m!); n!*polcoeff(A, n)}

%Y Cf. A219118.

%K sign

%O 0,3

%A _Paul D. Hanna_, Sep 19 2012

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