%I #9 May 07 2024 09:38:34

%S 1,0,1,0,2,-1,5,-7,15,-35,57,-155,262,-664,1297,-2910,6437,-13428,

%T 31461,-65137,152576,-325838,744223,-1649943,3685869,-8376976,

%U 18574146,-42579093,94912298,-217177891,489321856,-1114542791,2535640016,-5761630456,13184657747,-29989008137

%N Expansion of the series reversion of x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))), a continued fraction.

%C Reversion of g.f. (with constant term omitted) for A003823.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SeriesReversion.html">Series Reversion</a>

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F G.f. A(x) satisfies: A(x)/(1 + A(x)^2/(1 + A(x)^3/(1 + A(x)^4/(1 + A(x)^5/(1 + ...))))) = x.

%F a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 2.3794295463748306617... and c = 0.1900533719371157... - _Vaclav Kotesovec_, May 07 2024

%t Rest[CoefficientList[InverseSeries[Series[ContinuedFractionK[x^i, 1, {i, 1, 36}], {x, 0, 36}], x], x]]

%t Rest[CoefficientList[InverseSeries[Series[-1 + QPochhammer[x^2, x^5] QPochhammer[x^3, x^5]/(QPochhammer[x, x^5] QPochhammer[x^4, x^5]), {x, 0, 36}], x], x]]

%t (* Calculation of constant d: *) -1/r /. FindRoot[{1 + r == QPochhammer[s^2, s^5]*QPochhammer[s^3, s^5] / (QPochhammer[s, s^5]*QPochhammer[s^4, s^5]), 5*s^4*QPochhammer[s^3, s^5] * Derivative[0, 1][QPochhammer][s^2, s^5] + (1/s)* QPochhammer[s^2, s^5]*((1/Log[s^5])*QPochhammer[s^3, s^5] * (QPolyGamma[0, Log[s]/Log[s^5], s^5] - 2*QPolyGamma[0, Log[s^2]/Log[s^5], s^5] - 3*QPolyGamma[0, Log[s^3]/Log[s^5], s^5] + 4*QPolyGamma[0, Log[s^4]/Log[s^5], s^5]) - (5*s^5*QPochhammer[s^3, s^5] * Derivative[0, 1][QPochhammer][s, s^5])/ QPochhammer[s, s^5] + 5*s^5*Derivative[0, 1][QPochhammer][s^3, s^5] - (5*s^5*QPochhammer[s^3, s^5] * Derivative[0, 1][QPochhammer][s^4, s^5])/ QPochhammer[s^4, s^5]) == 0}, {r, -2/5}, {s, -2/ 3}, WorkingPrecision -> 60] (* _Vaclav Kotesovec_, May 07 2024 *)

%Y Cf. A003823.

%K sign

%O 1,5

%A _Ilya Gutkovskiy_, Aug 23 2017