OFFSET
0,4
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
E.g.f.: ( -LambertW(-x)*LambertW(x)/x^2 )^(1/(2*x)).
a(n) ~ c * n^(n-1), where c = -1/2*exp(1-exp(1))*(LambertW(exp(-1))^(-exp(1)/2)-LambertW(exp(-1))^(exp(1)/2)*exp(2*exp(1))) = 3.11403283... if n is even, and c = 1/2*exp(1-exp(1))*(LambertW(exp(-1))^(-exp(1)/2)+LambertW(exp(-1))^(exp(1)/2)*exp(2*exp(1))) = 4.13355253... if n is odd. - Vaclav Kotesovec, Jul 08 2013
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 17*x^3/3! + 65*x^4/4! + 1457*x^5/5! +...
such that A(x) = exp( Shw(x) ) where
Shw(x) = x + 16*x^3/3! + 1296*x^5/5! + 262144*x^7/7! + 100000000*x^9/9! +...+ (2*n+2)^(2*n)*x^(2*n+1)/(2*n+1)! +...
Related expansions:
Chw(x) = 1 + 3*x^2/2! + 125*x^4/4! + 16807*x^6/6! + 4782969*x^8/8! +...+ (2*n+1)^(2*n-1)*x^(2*n)/(2*n)! +...
where Chw(x) + Shw(x) = LambertW(-x)/(-x).
Also, A(x)^x is an even function:
A(x)^x = 1 + 2*x^2/2! + 76*x^4/4! + 9816*x^6/6! + 2731408*x^8/8! +...+ A215880(2*n)*x^(2*n)/(2*n)! +....
MATHEMATICA
max = 19; Shw[x_] := -(ProductLog[-x] + ProductLog[x])/(2*x); se = Series[Exp[Shw[x]] , {x, 0, max}]; (CoefficientList[se, x] // DeleteCases[#, 0] &)*Range[0, max]! (* Jean-François Alcover, Jun 24 2013 *)
PROG
(PARI) {a(n)=n!*polcoeff(exp(sum(m=0, n, (2*m+2)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n)), n)}
(PARI) {a(n)=local(LW=sum(m=1, n+2, -(-1)^m*m^(m-1)*x^m/m!)+x^4*O(x^n)); n!*polcoeff( (-subst(LW, x, -x)*LW/x^2)^(1/(2*x)), n)}
for(n=0, 21, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 02 2012
STATUS
approved