A307399 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} k*x^k*A(x)^k/(1 - x^k*A(x)^k).
1, 1, 4, 14, 60, 262, 1218, 5798, 28364, 141239, 714532, 3660098, 18949830, 98997082, 521218206, 2762807736, 14731968812, 78968221213, 425282844540, 2299997984844, 12485972925500, 68015653648096, 371666798915578, 2036765196573550, 11190993772943502, 61637787236407747
Offset: 0
Keywords
Examples
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 + ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
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] 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] terms = 26; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[DivisorSigma[1, k] x^k, {k, 1, terms}]), {x, 0, terms}], x], x] (* 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 *)
Formula
G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} sigma(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} sigma(k)*x^k)).
a(n) ~ c * d^n / n^(3/2), where d = A366072 = 5.84278321476352032847350429253643509033417800773284061845774243558820314... and c = 0.5552806478004840811027181339325620905324642078294... - Vaclav Kotesovec, Apr 07 2019