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Reversion of g.f. (with constant term omitted) for partition numbers.
(Formerly M1482)
11

%I M1482 #34 Jan 18 2024 06:23:18

%S 1,-2,5,-15,52,-200,825,-3565,15900,-72532,336539,-1582593,7524705,

%T -36111810,174695712,-851020367,4171156249,-20555470155,101787990805,

%U -506227992092,2527493643612,-12663916942984,63656297034920,-320914409885850,1622205233276889

%N Reversion of g.f. (with constant term omitted) for partition numbers.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vaclav Kotesovec, <a href="/A007312/b007312.txt">Table of n, a(n) for n = 1..1000</a>

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

%F From _Vaclav Kotesovec_, Nov 11 2017: (Start)

%F a(n) ~ -(-1)^n * c * d^n / n^(3/2), where

%F d = 5.379264118840884783404842050140885100801253519243086... and

%F c = 0.10697042824132534557642152089737206588353695053... (End)

%F G.f. A(x) satisfies: A(x) = 1 - (1/(1 + x)) * Product_{k>=2} 1/(1 - A(x)^k). - _Ilya Gutkovskiy_, Apr 23 2020

%p # Using function CompInv from A357588.

%p CompInv(25, n -> combinat:-numbpart(n)); # _Peter Luschny_, Oct 05 2022

%t nmax = 30; Rest[CoefficientList[InverseSeries[Series[Sum[PartitionsP[n]*x^n, {n, 1, nmax}], {x, 0, nmax}]], x]] (* _Vaclav Kotesovec_, Nov 11 2017 *)

%t Rest[CoefficientList[InverseSeries[Series[-1 + 1/QPochhammer[x],{x,0,30}],x],x]] (* _Vaclav Kotesovec_, Jan 18 2024 *)

%t (* Calculation of constant d: *) Chop[1/r /. FindRoot[{(1 + r)*QPochhammer[s, s] == 1, Log[1 - s] + QPolyGamma[0, 1, s] - (1 + r)*s*Log[s] * Derivative[0, 1][QPochhammer][s, s] == 0}, {r, -1/5}, {s, -1/2}, WorkingPrecision -> 70]] (* _Vaclav Kotesovec_, Jan 18 2024 *)

%Y Cf. A000041, A050393, A066398, A334315.

%K sign,easy

%O 1,2

%A _N. J. A. Sloane_, _Mira Bernstein_

%E Signs corrected Dec 24 2001