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 A007312 Reversion of g.f. (with constant term omitted) for partition numbers. (Formerly M1482) 11
 1, -2, 5, -15, 52, -200, 825, -3565, 15900, -72532, 336539, -1582593, 7524705, -36111810, 174695712, -851020367, 4171156249, -20555470155, 101787990805, -506227992092, 2527493643612, -12663916942984, 63656297034920, -320914409885850, 1622205233276889 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vaclav Kotesovec, Table of n, a(n) for n = 1..1000 Index entries for reversions of series FORMULA From Vaclav Kotesovec, Nov 11 2017: (Start) a(n) ~ -(-1)^n * c * d^n / n^(3/2), where d = 5.379264118840884783404842050140885100801253519243086... and c = 0.10697042824132534557642152089737206588353695053... (End) G.f. A(x) satisfies: A(x) = 1 - (1/(1 + x)) * Product_{k>=2} 1/(1 - A(x)^k). - Ilya Gutkovskiy, Apr 23 2020 MAPLE # Using function CompInv from A357588. CompInv(25, n -> combinat:-numbpart(n)); # Peter Luschny, Oct 05 2022 MATHEMATICA nmax = 30; Rest[CoefficientList[InverseSeries[Series[Sum[PartitionsP[n]*x^n, {n, 1, nmax}], {x, 0, nmax}]], x]] (* Vaclav Kotesovec, Nov 11 2017 *) Rest[CoefficientList[InverseSeries[Series[-1 + 1/QPochhammer[x], {x, 0, 30}], x], x]] (* Vaclav Kotesovec, Jan 18 2024 *) (* 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 *) CROSSREFS Cf. A000041, A050393, A066398, A334315. Sequence in context: A369443 A369398 A370798 * A007296 A279558 A224071 Adjacent sequences: A007309 A007310 A007311 * A007313 A007314 A007315 KEYWORD sign,easy AUTHOR N. J. A. Sloane, Mira Bernstein EXTENSIONS Signs corrected Dec 24 2001 STATUS approved

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Last modified June 22 15:19 EDT 2024. Contains 373587 sequences. (Running on oeis4.)