OFFSET
0,2
FORMULA
E.g.f. A(x) satisfies:
(1) A(x) = (1/x) * Series_Reversion( 3*x - 2*x*exp(x) ).
(2) A(x) = 1 + (1/x) * Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^n / n!.
(3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) = A259063(n+1) / (n+1). - Vaclav Kotesovec, Jun 19 2015
a(n) ~ (c/3)^(n+1) * n^(n-1) / (sqrt(c+1) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(3*exp(1)/2). - Vaclav Kotesovec, Jun 19 2015
a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^k * (n+k)! * Stirling2(n,k). - Seiichi Manyama, Mar 06 2024
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 302*x^3/3! + 7562*x^4/4! + 253542*x^5/5! +...
where A(3*x - 2*x*exp(x)) = 1/(3 - 2*exp(x)).
MATHEMATICA
CoefficientList[1/x*InverseSeries[Series[3*x - 2*x*E^x, {x, 0, 21}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 19 2015 *)
PROG
(PARI) {a(n) = local(A=1); A = (1/x)*serreverse(3*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=1); A = 1 + (1/x)*sum(m=1, n+1, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^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=1+x+x*O(x^n)); A = exp(sum(m=1, n+1, 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=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 18 2015
STATUS
approved