OFFSET
0,3
COMMENTS
Compare the e.g.f. to the identity: Sum_{n>=0} n^n * exp(-n*x) * x^n/n! = 1/(1-x).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
FORMULA
E.g.f.: 1 + (1/2)*x/(1-x) - (1/2)*LambertW(-x*exp(x))/(1 + LambertW(-x*exp(x))).
a(n) ~ n^n/(2*sqrt(1+LambertW(exp(-1)))*exp(n)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Jul 08 2013
a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^n * binomial(n,2*k). - Seiichi Manyama, Feb 15 2023
EXAMPLE
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 30*x^3/3! + 352*x^4/4! + 5560*x^5/5! +...
where
A(x) = 1 + 1^1*x*cosh(1*x) + 2^2*cosh(2*x)*x^2/2! + 3^3*cosh(3*x)*x^3/3! + 4^4*cosh(4*x)*x^4/4! + 5^5*cosh(5*x)*x^5/5! +...
MATHEMATICA
CoefficientList[Series[1 + 1/2*x/(1-x) - 1/2*LambertW[-x*E^x]/(1 + LambertW[-x*E^x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)
PROG
(PARI) a(n)=n!*polcoeff(sum(k=0, n, k^k*cosh(k*x +x*O(x^n))*x^k/k!), n)
for(n=0, 30, print1(a(n), ", "))
(PARI) LambertW(x, N)=sum(n=1, N, (-n)^(n-1)*x^n/n!)
{a(n)=local(X=x+x*O(x^n)); n!*polcoeff(1 + (1/2)*x/(1-X) - (1/2)*LambertW(-x*exp(X), n)/(1 + LambertW(-x*exp(X), n)), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 08 2013
(PARI) a(n) = sum(k=0, n\2, (n-2*k)^n*binomial(n, 2*k)); \\ Seiichi Manyama, Feb 15 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 27 2012
EXTENSIONS
E.g.f. in Formula section corrected by Paul D. Hanna, Jul 08 2013, error noted by Vaclav Kotesovec.
STATUS
approved