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%I #12 Sep 27 2023 12:08:06
%S 1,1,1,3,8,18,50,150,429,1258,3835,11740,36148,112856,355318,1124582,
%T 3582186,11477162,36939043,119387415,387393424,1261422550,4120343870,
%U 13498085604,44337516318,145993301239,481812344551,1593439356575,5280074015618,17528034861180
%N G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^k*A(x)^k/(1 + x^k*A(x)^k).
%H Vaclav Kotesovec, <a href="/A307397/b307397.txt">Table of n, a(n) for n = 0..1835</a>
%F G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} A048272(k)*x^k*A(x)^k.
%F G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} A048272(k)*x^k)).
%F a(n) ~ c * d^n / n^(3/2), where d = 3.494393309755265712092948136162079492013891957890570364999827377394989... and c = 0.4966863488644340281558816065879601408815044380350316600850227488... - _Vaclav Kotesovec_, Sep 27 2023
%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 8*x^4 + 18 x^5 + 50*x^6 + 150*x^7 + 429*x^8 + 1258*x^9 + 3835*x^10 + ...
%t terms = 30; A[_] = 0; Do[A[x_] = 1 + Sum[x^k A[x]^k/ (1 + x^k A[x]^k), {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
%t terms = 30; A[_] = 0; Do[A[x_] = 1 + Sum[Sum[(-1)^(d + 1), {d, Divisors[k]}] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
%t terms = 30; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[Sum[(-1)^(d + 1), {d, Divisors[k]}] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
%t (* Calculation of constant d: *) val = r /. FindRoot[{Log[1 - r*s] + QPolyGamma[0, 1 - I*Pi/Log[r*s], r*s] == (1 - s)*Log[r*s], (1 - s)/s - I*Pi*QPolyGamma[1, 1 - I*Pi/Log[r*s], r*s] / (s*Log[r*s]^2) + r*(1/(1 - r*s) - Derivative[0, 0, 1][QPolyGamma][0, 1 - I*Pi/Log[r*s], r*s]) == Log[r*s]}, {r, 1/3}, {s, 2}, WorkingPrecision -> 100]; N[1/Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* _Vaclav Kotesovec_, Sep 27 2023 *)
%Y Cf. A048272, A190790, A192206, A307399, A307401.
%K nonn
%O 0,4
%A _Ilya Gutkovskiy_, Apr 07 2019
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