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A178451 Coefficients in series expansion of 1/j_inv, where j_inv (A091406) is the reversion of the j-function. 4

%I #21 Mar 07 2018 08:52:55

%S 1,-744,-196884,-167975456,-180592706130,-217940004309744,

%T -282054965806724344,-382591095354251539392,-536797252082856840544683,

%U -772598111838972001258770120,-1134346327935015067651297762308,-1692324738742597705005194275401888,-2558136060792026773012451913035887538,-3909566534059719280565543662082528637552,-6030806348626044568366137322595811547663800,-9377648421379464305085605549750143357652168640,-14683413510495912973021347501907744913788055440950

%N Coefficients in series expansion of 1/j_inv, where j_inv (A091406) is the reversion of the j-function.

%H Vaclav Kotesovec, <a href="/A178451/b178451.txt">Table of n, a(n) for n = -1..300</a>

%H Y.-H. He and V. Jejjala, <a href="http://arXiv.org/abs/hep-th/0307293">Modular Matrix Models</a>. See Eq. 72.

%F a(n) ~ c * 1728^n / n^(3/2), where c = -2 * exp(2*Pi) * Gamma(3/4)^4 / (sqrt(3) * Pi^(3/2)) = -250.3989185574412282530281713739868122541444992745630952... - _Vaclav Kotesovec_, Jul 03 2017, updated Mar 07 2018

%t nmax = 20; s1 = 1728*Series[KleinInvariantJ[t], {t, 0, 2*nmax}] /.t -> -2*I*(Pi/Log[q]); s2 = Normal[InverseSeries[Series[s1, {q, 0, nmax}], j]] /.j -> 1/x; CoefficientList[Series[x/s2, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 03 2017 after _Jean-François Alcover_ *)

%o (PARI)

%o x = 'x+O('x^50);

%o A=x*(eta(x^2)/eta(x))^24;

%o r=serreverse(A/(1+256*A)^3);

%o Vec( 1/r ) /* show terms */

%Y Cf. A000521, A091406, A066396.

%K sign,easy

%O -1,2

%A _N. J. A. Sloane_, Dec 22 2010

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)