A219118 E.g.f.: Sum_{n>=0} log(1 + x*exp(2*n*x))^n / n!.

1, 1, 4, 24, 296, 4280, 79032, 1978368, 57803776, 1949421888, 78945675200, 3678493774560, 190462000632576, 11112878148649472, 730288018660087552, 52727783838765181440, 4148774572438335014912, 358041540338404096024576, 33700760914469383117799424

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

E.g.f. satisfies the identities:

(1) Sum_{n>=0} binomial(exp(2*n*x),n) * x^n.

(2) Sum_{n>=0} [Product_{k=0..n-1} (exp(2*n*x) - k)] * x^n/n!.

(3) Sum_{n>=0} x^n * Sum_{k=0..n} Stirling1(n,k) * exp(2*n*k*x) / n!.

Example

 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 24*x^3/3! + 296*x^4/4! + 4280*x^5/5! +...
where the e.g.f. equals the series:
(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! +...
(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 +...
(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! +...

Prog

 (PARI) {a(n)=n!*polcoeff(sum(m=0, n, log(1+x*exp(2*m*x+x*O(x^n)))^m/m!), n)}
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, binomial(exp(2*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(2*m*x +x*O(x^n)) - k)) * x^m/m!), 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(2*m*k*x+x*O(x^n)))*x^m/m!); n!*polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Crossrefs

Cf. A216839.

Sequence in context: A009104 A255440 A052727 * A005756 A206239 A374386

Adjacent sequences: A219115 A219116 A219117 * A219119 A219120 A219121

Keyword

nonn

Author

Paul D. Hanna, Nov 12 2012

Status

approved