OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..329
FORMULA
O.g.f.: x * Sum_{n>=0} 2^n / (3 - n*x)^(n+1).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^(n-1) / n! ).
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
EXAMPLE
E.g.f.: A(x) = x + 4*x^2/2! + 54*x^3/3! + 1208*x^4/4! + 37810*x^5/5! + ...
where A(3*x - 2*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 4*x^2 + 54*x^3 + 1208*x^4 + 37810*x^5 + 1521252*x^6 + ...
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 +...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[3*x - 2*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(3*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*(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, 2^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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 17 2015
STATUS
approved