A196198 E.g.f. satisfies A(x) = exp(x/A(-x)).

1, 1, 3, 4, -19, -64, 1207, 5440, -164071, -954368, 39943691, 284754944, -15250391099, -128749666304, 8402599565375, 81978198409216, -6309988001033167, -69853770233675776, 6194681665486634899, 76717804389440684032

Offset: 0

Author

Paul D. Hanna, Sep 30 2011

Keywords

• sign

Examples

			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! +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..406

Crossrefs

Cf. A141369.

Programs

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 *)

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)}

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