A259064 E.g.f.: Series_Reversion( 4*x - 3*x*exp(x) ).

1, 6, 117, 3792, 172005, 10030248, 714843885, 60207412128, 5850995291397, 644410711219800, 79322681596610661, 10791841135527454896, 1608054016025580893445, 260445389091217967677992, 45557042043723212142506205, 8559094926999510089793332544, 1718950045690606262911636792677

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..315

Formula

O.g.f.: x * Sum_{n>=0} 3^n / (4 - n*x)^(n+1).

E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * (exp(x)-1)^n * x^n / n!.

E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * (exp(x)-1)^n * x^(n-1) / n! ).

a(n) ~ (c/(4*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(4*exp(1)/3). - Vaclav Kotesovec, Jun 19 2015

Example

E.g.f.: A(x) = x + 6*x^2/2! + 117*x^3/3! + 3792*x^4/4! + 172005*x^5/5! +...
where A(4*x - 3*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 6*x^2 + 117*x^3 + 3792*x^4 + 172005*x^5 + 10030248*x^6 +...
where F(x)/x = 1/4 + 3/(4-x)^2 + 3^2/(4-2*x)^3 + 3^3/(4-3*x)^4 + 3^4/(4-4*x)^5 +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[4*x - 3*x*E^x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)

Prog

(PARI) {a(n) = local(A=x); A = serreverse(4*x - 3*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, 3^m*(exp(x+x*O(x^n))-1)^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, 3^m*(exp(x+x*O(x^n))-1)^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A053492, A259063, A259065, A259066.

Sequence in context: A362765 A024275 A287652 * A300733 A332627 A100070

Adjacent sequences: A259061 A259062 A259063 * A259065 A259066 A259067

Keyword

nonn

Author

Paul D. Hanna, Jun 17 2015

Status

approved