A259062 E.g.f.: Series_Reversion( -x + 2*x*exp(-x) ).

1, 4, 42, 728, 17630, 548532, 20852370, 936655792, 48540537702, 2850727359500, 187107038833946, 13572973331551944, 1078343465147156910, 93119965280416893028, 8684514946963752624930, 869915871265946242868576, 93146889134541855185069942, 10617155946603647157142073916

Offset: 1

Author

Paul D. Hanna, Jun 18 2015

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 728*x^4/4! + 17630*x^5/5! +...
where A(-x + 2*x*exp(-x)) = x.
		

Crossrefs

Cf. A258872, A053492, A259063, A259064, A259065, A259066.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[-x + 2*x*E^(-x), {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)

PARI

{a(n) = local(A=x); A = serreverse(-x + 2*x*exp(-x +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x*O(x^n)))^m*x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x*O(x^n)))^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^n / n!.

E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^(n-1) / n! ).

a(n) ~ (1-c) * n^(n-1) / (sqrt(1+c) * (c + 1/c - 2)^n * exp(n)), where c = LambertW(exp(1)/2) = 0.685076942154593946... . - Vaclav Kotesovec, Jun 19 2015