This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A258923 #24 Mar 06 2024 08:00:16
%S A258923 1,3,39,948,34401,1671708,102120555,7525926516,650110587933,
%T A258923 64441071121980,7211152872419151,899320094627287908,
%U A258923 123696462771198530265,18603242077944140548428,3037136136248214142833747,534943432937469380612083284,101114708570035662524213928981,20416341060201627868414787791068
%N A258923 E.g.f. satisfies A(x) = 1/(4 - 3*exp(x*A(x))).
%F A258923 E.g.f. A(x) satisfies:
%F A258923 (1) A(x) = (1/x) * Series_Reversion( 4*x - 3*x*exp(x) ).
%F A258923 (2) A(x) = 1 + (1/x) * Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * (exp(x)-1)^n * x^n / n!.
%F A258923 (3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * (exp(x)-1)^n * x^(n-1) / n! ).
%F A258923 a(n) = A259064(n+1) / (n+1). - _Vaclav Kotesovec_, Jun 19 2015
%F A258923 a(n) ~ (c/4)^(n+1) * n^(n-1) / (sqrt(c+1) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(4*exp(1)/3). - _Vaclav Kotesovec_, Jun 19 2015
%F A258923 a(n) = (1/(n+1)!) * Sum_{k=0..n} 3^k * (n+k)! * Stirling2(n,k). - _Seiichi Manyama_, Mar 06 2024
%e A258923 E.g.f.: A(x) = 1 + 3*x + 39*x^2/2! + 948*x^3/3! + 34401*x^4/4! + 1671708*x^5/5! +...
%e A258923 where A(4*x - 3*x*exp(x)) = 1/(4 - 3*exp(x)).
%t A258923 CoefficientList[1/x*InverseSeries[Series[4*x - 3*x*E^x, {x, 0, 21}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jun 19 2015 *)
%o A258923 (PARI) {a(n) = local(A=1); A = (1/x)*serreverse(4*x - 3*x*exp(x +x^2*O(x^n) )); n!*polcoeff(A, n)}
%o A258923 for(n=0, 20, print1(a(n), ", "))
%o A258923 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A258923 {a(n)=local(A=1); A = 1 + (1/x)*sum(m=1, n+1, Dx(m-1, 3^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
%o A258923 for(n=0, 25, print1(a(n), ", "))
%o A258923 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A258923 {a(n)=local(A=1+x+x*O(x^n)); A = exp(sum(m=1, n+1, 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 A258923 for(n=0, 25, print1(a(n), ", "))
%Y A258923 Cf. A259064, A052894, A258922.
%K A258923 nonn
%O A258923 0,2
%A A258923 _Paul D. Hanna_, Jun 18 2015