OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..406
FORMULA
a(n) = Sum_{k=0..n-1} binomial(n,k) * (n-k)^k * (-k+1)^(n-k-1) for n>0 with a(0)=1.
E.g.f. satisfies:
_ A(x) = exp(x*exp(x/A(x))).
_ A(x) = exp(x* exp(x*exp(-x*exp(x*exp(-x*exp(x*exp(-x*...))))))).
_ A(x) = exp(x*B(x)) where B(x) = exp(x/B(x)) is the e.g.f. of A141369.
E.g.f. satisfies: x/exp(-x/A(x)) = log(A(x)). - Vaclav Kotesovec, Feb 26 2014
|a(n)| ~ c * n! / (n^(3/2) * r^n), where r = 0.5098636055230131449434409623392631606695606770070519241... is the root of the equation r*exp(1/LambertW(-I/r))/I = LambertW(-I/r), and c = 0.385745347287849929987791864025522098993432068... if n is even, and c = 0.12921599603996711137996765405025929272341118... if n is odd. - Vaclav Kotesovec, Feb 26 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 4*x^3/3! - 19*x^4/4! - 64*x^5/5! +...
where log(A(x)) = x/A(-x) begins:
x/A(-x) = x + 2*x^2/2! - 3*x^3/3! - 32*x^4/4! + 105*x^5/5! + 2016*x^6/6! - 10115*x^7/7! - 282624*x^8/8! +...+ n*A141369(n-1)*x^n/n! +...
MATHEMATICA
Flatten[{1, 1, 3, Table[Sum[Binomial[n, k]*(n-k)^k*(-k+1)^(n-k-1), {k, 0, n-1}], {n, 3, 20}]}] (* Vaclav Kotesovec, Feb 26 2014 *)
PROG
(PARI) {a(n)=if(n==0, 1, sum(k=0, n-1, binomial(n, k)*(n-k)^k*(-k+1)^(n-k-1)))}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(x/subst(A, x, -x+x*O(x^n)))); n!*polcoeff(A, n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Sep 30 2011
STATUS
approved