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A291377
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Expansion of the series reversion of x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))), a continued fraction.
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1
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1, 0, 1, 0, 2, -1, 5, -7, 15, -35, 57, -155, 262, -664, 1297, -2910, 6437, -13428, 31461, -65137, 152576, -325838, 744223, -1649943, 3685869, -8376976, 18574146, -42579093, 94912298, -217177891, 489321856, -1114542791, 2535640016, -5761630456, 13184657747, -29989008137
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OFFSET
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1,5
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COMMENTS
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Reversion of g.f. (with constant term omitted) for A003823.
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LINKS
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FORMULA
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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.
a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 2.3794295463748306617... and c = 0.1900533719371157... - Vaclav Kotesovec, May 07 2024
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MATHEMATICA
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Rest[CoefficientList[InverseSeries[Series[ContinuedFractionK[x^i, 1, {i, 1, 36}], {x, 0, 36}], x], x]]
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]]
(* 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 *)
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CROSSREFS
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KEYWORD
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sign,changed
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AUTHOR
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STATUS
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approved
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