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 A385502 #18 Jul 01 2025 08:48:42
%S A385502 1,1,1,3,25,205,2001,25991,394353,6718041,130319745,2833146987,
%T A385502 67767170505,1772434086501,50392083769041,1546052750636655,
%U A385502 50905035315373281,1790951445870568113,67050161599822764417,2661363261252799648083,111637709182606749500025
%N A385502 E.g.f. A(x) satisfies A(x) = exp( arctanh(x * A(x)) / A(x) ).
%F A385502 E.g.f. A(x) satisfies A(x) = ( (1+x*A(x))/(1-x*A(x)) )^(1/(2*A(x))).
%F A385502 a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * A111594(n,k).
%F A385502 a(n) ~ s * sqrt((-1 + r^2*s^2)/(1 - 2*r^2*(1 + r)*s^2 + r^4*s^4)) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.4422573061236400123439455590007131605377062990202... and s = 1.93686591146053883124948614770176661449449740697... are the roots of the system of equations ((1 + r*s)/(1 - r*s))^(1/2/s) = s, 2*r*s + (-1 + r^2*s^2)*(log((1 + r*s)/(1 - r*s)) + 2*s) = 0. - _Vaclav Kotesovec_, Jul 01 2025
%t A385502 terms = 21; A[_] = 1; Do[A[x_] = Exp[ArcTanh[x*A[x]]/A[x]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* _Stefano Spezia_, Jul 01 2025 *)
%o A385502 (PARI) a111594(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(atanh(x)^k/k!, n);
%o A385502 a(n) = sum(k=0, n, (n-k+1)^(k-1)*a111594(n, k));
%Y A385502 Cf. A385500, A385501.
%Y A385502 Cf. A111594, A385428.
%K A385502 nonn
%O A385502 0,4
%A A385502 _Seiichi Manyama_, Jul 01 2025