A258872 E.g.f.: (1/x) * Series_Reversion( -x + 2*x*exp(-x) ).

1, 2, 14, 182, 3526, 91422, 2978910, 117081974, 5393393078, 285072735950, 17009730803086, 1131081110962662, 82949497319012070, 6651426091458349502, 578967663130916841662, 54369741954121640179286, 5479228772620109128533526, 589841997033535953174559662

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

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

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

a(n) = A259062(n+1) / (n+1). - Vaclav Kotesovec, Jun 19 2015

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

Example

E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 182*x^3/3! + 3526*x^4/4! + 91422*x^5/5! +...

Mathematica

CoefficientList[1/x*InverseSeries[Series[-x + 2*x*E^(-x), {x, 0, 21}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 19 2015 *)

Prog

(PARI) {a(n) = local(A=x); A = (1/x)*serreverse(-x + 2*x*exp(-x +x^2*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 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 = 1 + (1/x)*sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x^2*O(x^n)))^m*x^m/m!)); n!*polcoeff(A, n)}
for(n=0, 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 = exp(sum(m=1, n+1, 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=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A259062.

Sequence in context: A370054 A210097 A230991 * A372246 A000807 A191565

Adjacent sequences: A258869 A258870 A258871 * A258873 A258874 A258875

Keyword

nonn

Author

Paul D. Hanna, Jun 18 2015

Status

approved