A258922 - OEIS

%I #18 Mar 06 2024 08:00:21

%S 1,2,18,302,7562,253542,10685794,543309230,32378850042,2214215333750,

%T 170939286647570,14707184259036414,1395561779648175274,

%U 144795755972202587462,16308198003201872476866,1981633767850818093910094,258406311809937562215099482,35994776359231593721760238102

%N E.g.f. satisfies A(x) = 1/(3 - 2*exp(x*A(x))).

%F E.g.f. A(x) satisfies:

%F (1) A(x) = (1/x) * Series_Reversion( 3*x - 2*x*exp(x) ).

%F (2) A(x) = 1 + (1/x) * Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^n / n!.

%F (3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^(n-1) / n! ).

%F a(n) = A259063(n+1) / (n+1). - _Vaclav Kotesovec_, Jun 19 2015

%F 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

%F a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^k * (n+k)! * Stirling2(n,k). - _Seiichi Manyama_, Mar 06 2024

%e E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 302*x^3/3! + 7562*x^4/4! + 253542*x^5/5! +...

%e where A(3*x - 2*x*exp(x)) = 1/(3 - 2*exp(x)).

%t CoefficientList[1/x*InverseSeries[Series[3*x - 2*x*E^x, {x, 0, 21}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jun 19 2015 *)

%o (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)}

%o for(n=0, 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=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)}

%o for(n=0, 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=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)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A259063, A052894, A258923.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 18 2015