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A007312 Reversion of g.f. (with constant term omitted) for partition numbers.

%I A007312 M1482 #34 Jan 18 2024 06:23:18
%S A007312 1,-2,5,-15,52,-200,825,-3565,15900,-72532,336539,-1582593,7524705,
%T A007312 -36111810,174695712,-851020367,4171156249,-20555470155,101787990805,
%U A007312 -506227992092,2527493643612,-12663916942984,63656297034920,-320914409885850,1622205233276889
%N A007312 Reversion of g.f. (with constant term omitted) for partition numbers.
%D A007312 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007312 Vaclav Kotesovec, <a href="/A007312/b007312.txt">Table of n, a(n) for n = 1..1000</a>
%H A007312 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A007312 From _Vaclav Kotesovec_, Nov 11 2017: (Start)
%F A007312 a(n) ~ -(-1)^n * c * d^n / n^(3/2), where
%F A007312 d = 5.379264118840884783404842050140885100801253519243086... and
%F A007312 c = 0.10697042824132534557642152089737206588353695053... (End)
%F A007312 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 A007312 # Using function CompInv from A357588.
%p A007312 CompInv(25, n -> combinat:-numbpart(n)); # _Peter Luschny_, Oct 05 2022
%t A007312 nmax = 30; Rest[CoefficientList[InverseSeries[Series[Sum[PartitionsP[n]*x^n, {n, 1, nmax}], {x, 0, nmax}]], x]] (* _Vaclav Kotesovec_, Nov 11 2017 *)
%t A007312 Rest[CoefficientList[InverseSeries[Series[-1 + 1/QPochhammer[x],{x,0,30}],x],x]] (* _Vaclav Kotesovec_, Jan 18 2024 *)
%t A007312 (* 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 A007312 Cf. A000041, A050393, A066398, A334315.
%K A007312 sign,easy
%O A007312 1,2
%A A007312 _N. J. A. Sloane_, _Mira Bernstein_
%E A007312 Signs corrected Dec 24 2001