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A257544 Expansion of e.g.f. 1/(1 - x/(1 - (x^2/2!)/(1 - (x^3/3!)/(1 - (x^4/4!)/(1 - (x^5/5!)/(1 -... (x^n/n!)/(1 -...))))))), a continued fraction.

%I A257544 #14 May 18 2025 07:55:53
%S A257544 1,1,2,9,48,330,2760,26670,295680,3686760,51067800,778239000,
%T A257544 12937478400,232997965200,4518981066600,93905637625800,
%U A257544 2081477759961600,49020847858382400,1222400311155223200,32175639432544608000,891491725439101987200,25935626681521134177600,790459388855441635730400
%N A257544 Expansion of e.g.f. 1/(1 - x/(1 - (x^2/2!)/(1 - (x^3/3!)/(1 - (x^4/4!)/(1 - (x^5/5!)/(1 -... (x^n/n!)/(1 -...))))))), a continued fraction.
%C A257544 Limit n * a(n)/a(n+1) = 0.721838205529056...
%H A257544 Vaclav Kotesovec, <a href="/A257544/b257544.txt">Table of n, a(n) for n = 0..300</a>
%F A257544 a(n) ~ c * d^n * n!, where d = 1.385351997636466377154331548358490992212171811742252828163104808424069377... and c = 0.54049290018070557137173379570192477214723097220914603426642758802700306... - _Vaclav Kotesovec_, Aug 24 2017
%e A257544 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 330*x^5/5! +...
%e A257544 where
%e A257544 A(x) = 1 / (1 - x / (1 - (x^2/2!) / (1 - (x^3/3!) / (1 - (x^4/4!) / (1 - (x^5/5!) / (1 - (x^6/6!) / (1 - (x^7/7!) / (1 -...)))))))).
%t A257544 nmax = 30; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^Range[nmax + 1]/Range[nmax + 1]!]], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Aug 24 2017 *)
%o A257544 (PARI) {a(n)=local(CF=1+x*O(x^n), M=n+1); for(k=0, M, CF=1/(1-x^(M-k+1)/(M-k+1)!*CF)); n!*polcoeff(CF, n, x)}
%o A257544 for(n=0, 25, print1(a(n), ", "))
%K A257544 nonn
%O A257544 0,3
%A A257544 _Paul D. Hanna_, Apr 28 2015