A259064 - OEIS

%I #18 Nov 16 2017 02:56:08

%S 1,6,117,3792,172005,10030248,714843885,60207412128,5850995291397,

%T 644410711219800,79322681596610661,10791841135527454896,

%U 1608054016025580893445,260445389091217967677992,45557042043723212142506205,8559094926999510089793332544,1718950045690606262911636792677

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

%H G. C. Greubel, <a href="/A259064/b259064.txt">Table of n, a(n) for n = 1..315</a>

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

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

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

%F 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

%e E.g.f.: A(x) = x + 6*x^2/2! + 117*x^3/3! + 3792*x^4/4! + 172005*x^5/5! +...

%e where A(4*x - 3*x*exp(x)) = x.

%e Also we have the related infinite series.

%e O.g.f.: F(x) = x + 6*x^2 + 117*x^3 + 3792*x^4 + 172005*x^5 + 10030248*x^6 +...

%e 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 +...

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

%o (PARI) {a(n) = local(A=x); A = serreverse(4*x - 3*x*exp(x +x*O(x^n) )); n!*polcoeff(A,n)}

%o for(n=1,20,print1(a(n),", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {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)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A053492, A259063, A259065, A259066.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 17 2015