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%I #13 Oct 02 2023 08:18:46
%S 1,1,4,14,60,262,1218,5798,28364,141239,714532,3660098,18949830,
%T 98997082,521218206,2762807736,14731968812,78968221213,425282844540,
%U 2299997984844,12485972925500,68015653648096,371666798915578,2036765196573550,11190993772943502,61637787236407747
%N G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} k*x^k*A(x)^k/(1 - x^k*A(x)^k).
%H Vaclav Kotesovec, <a href="/A307399/b307399.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} sigma(k)*x^k*A(x)^k.
%F G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} sigma(k)*x^k)).
%F a(n) ~ c * d^n / n^(3/2), where d = A366072 = 5.84278321476352032847350429253643509033417800773284061845774243558820314... and c = 0.5552806478004840811027181339325620905324642078294... - _Vaclav Kotesovec_, Apr 07 2019
%e G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 60*x^4 + 262*x^5 + 1218*x^6 + 5798*x^7 + 28364*x^8 + 141239*x^9 + 714532*x^10 + ...
%t terms = 26; A[_] = 0; Do[A[x_] = 1 + Sum[k 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 = 26; A[_] = 0; Do[A[x_] = 1 + Sum[DivisorSigma[1, k] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
%t terms = 26; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[DivisorSigma[1, k] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
%t (* Calculation of constant d: *) val = r /. FindRoot[{1 + (Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / Log[r*s] == s + r*s*Derivative[0, 1][QPochhammer][r*s, r*s]/ QPochhammer[r*s, r*s], r^2*s*Derivative[0, 1][QPochhammer][r*s, r*s]^2/ QPochhammer[r*s, r*s]^2 + 1/(s*(-1 + r*s)*Log[r*s]^2) * (2*r*s*Log[r*s] + s*Log[r*s]^2 - r*s^2*Log[r*s]^2 + 2*Log[1 - r*s] - 2*r*s*Log[1 - r*s] + (2 - 2*r*s)*QPolyGamma[0, 1, r*s] + (1 - r*s)*QPolyGamma[1, 1, r*s] - 2*r*s*Log[r*s] * Derivative[0, 0, 1][QPolyGamma][0, 1, r*s] + 2*r^2*s^2*Log[r*s]*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]) == (r*(Derivative[0, 1][QPochhammer][r*s, r*s] + r*s*Derivative[0, 2][QPochhammer][r*s, r*s]))/QPochhammer[r*s, r*s]}, {r, 1/6}, {s, 2}, WorkingPrecision -> 40] // Quiet; N[ 1/Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* _Vaclav Kotesovec_, Oct 02 2023 *)
%Y Cf. A000203, A192206, A307397, A307401.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 07 2019
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