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

1, 1, 2, 9, 64, 515, 6126, 87332, 1408352, 28357821, 656029450, 16616305354, 486491747952, 16101080888763, 572203757798414, 22348109637703800, 973262507935361632, 45353465796372720729, 2238286744709428606866, 120361307277708751011502

Offset

0,3

Comments

Note that a(32)-a(42), a(57)-a(69), ... are negative, see b-file. - Vaclav Kotesovec, Nov 05 2014

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Formula

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

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

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

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 515*x^5/5! +...
where the g.f. satisfies the identities:
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! +...
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 +...
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! +...

Prog

(PARI) {a(n)=n!*polcoeff(sum(m=0, n, log(1+x*exp(m*x+x*O(x^n)))^m/m!), n)}
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, binomial(exp(m*x+x*O(x^n)), m)*x^m), n)}
(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)}
for(n=0, 31, print1(a(n), ", "))
(PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{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)}

Crossrefs

Cf. A219118.

Sequence in context: A076944 A074181 A052513 * A024720 A289717 A094100

Adjacent sequences: A216836 A216837 A216838 * A216840 A216841 A216842

Keyword

sign

Author

Paul D. Hanna, Sep 19 2012

Status

approved