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A259063 E.g.f.: Series_Reversion( 3*x - 2*x*exp(x) ).

%I A259063 #20 Nov 16 2017 02:55:48
%S A259063 1,4,54,1208,37810,1521252,74800558,4346473840,291409650378,
%T A259063 22142153337500,1880332153123270,176486211108436968,
%U A259063 18142303135426278562,2027140583610836224468,244622970048028087152990,31706140285613089502561504,4392907300768938557656691194,647905974466168686991684285836
%N A259063 E.g.f.: Series_Reversion( 3*x - 2*x*exp(x) ).
%H A259063 G. C. Greubel, <a href="/A259063/b259063.txt">Table of n, a(n) for n = 1..329</a>
%F A259063 O.g.f.: x * Sum_{n>=0}  2^n / (3 - n*x)^(n+1).
%F A259063 E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^n / n!.
%F A259063 E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^(n-1) / n! ).
%F A259063 a(n) ~ (c/(3*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(3*exp(1)/2). - _Vaclav Kotesovec_, Jun 19 2015
%e A259063 E.g.f.: A(x) = x + 4*x^2/2! + 54*x^3/3! + 1208*x^4/4! + 37810*x^5/5! + ...
%e A259063 where A(3*x - 2*x*exp(x)) = x.
%e A259063 Also we have the related infinite series.
%e A259063 O.g.f.: F(x) = x + 4*x^2 + 54*x^3 + 1208*x^4 + 37810*x^5 + 1521252*x^6 + ...
%e A259063 where F(x)/x = 1/3 + 2/(3-x)^2 + 2^2/(3-2*x)^3 + 2^3/(3-3*x)^4 + 2^4/(3-4*x)^5 +...
%t A259063 Rest[CoefficientList[InverseSeries[Series[3*x - 2*x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jun 19 2015 *)
%o A259063 (PARI) {a(n) = local(A=x); A = serreverse(3*x - 2*x*exp(x +x*O(x^n) )); n!*polcoeff(A,n)}
%o A259063 for(n=1,20,print1(a(n),", "))
%o A259063 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A259063 {a(n)=local(A=x); A = x + sum(m=1, n, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
%o A259063 for(n=1, 25, print1(a(n), ", "))
%o A259063 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A259063 {a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o A259063 for(n=1, 25, print1(a(n), ", "))
%Y A259063 Cf. A053492, A259064, A259065, A259066, A259062, A258922.
%K A259063 nonn
%O A259063 1,2
%A A259063 _Paul D. Hanna_, Jun 17 2015