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A034318 McKay-Thompson series of class 13A for the Monster group with a(0) = -2. 5

%I #31 Feb 19 2018 18:07:39

%S 1,-2,12,28,66,132,258,468,843,1428,2406,3900,6253,9780,15144,22980,

%T 34599,51300,75430,109584,158052,225676,320082,450216,629329,873444,

%U 1205514,1653364,2256087,3061620,4135280,5557980,7438170,9910132,13151568,17382756,22891391

%N McKay-Thompson series of class 13A for the Monster group with a(0) = -2.

%H Vaclav Kotesovec, <a href="/A034318/b034318.txt">Table of n, a(n) for n = -1..5000</a>

%H I. Chen and N. Yui, <a href="http://www.math.sfu.ca/~ichen/pub.html">Singular values of Thompson series</a>. In Groups, difference sets and the Monster (Columbus, OH, 1993), pp. 255-326, Ohio State University Mathematics Research Institute Publications, 4, de Gruyter, Berlin, 1996.

%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F Expansion of Hauptmodul for Gamma_0(13)+.

%F Expansion of (eta(q) / eta(q^13))^2 + 13*(eta(q^13) / eta(q))^2 in powers of q. - _Michael Somos_, Jul 05 2012

%F G.f.: (1/x) * Product_{k>0} ((1 - x^k) / (1 - x^(13*k)))^2 + 13*x * Product_{k>0} ((1 - x^(13*k)) / (1 - x^k))^2. - _Michael Somos_, Jul 05 2012

%F a(n) = A034319(n) unless n=0. - _Michael Somos_, Jul 05 2012

%F a(n) = A133099(n) + 13 * A121597(n). - _Michael Somos_, Jul 05 2012

%F a(n) ~ exp(4*Pi*sqrt(n/13)) / (sqrt(2) * 13^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2017

%e G.f. = 1/q - 2 + 12*q + 28*q^2 + 66*q^3 + 132*q^4 + 258*q^5 + 468*q^6 + ...

%t QP = QPochhammer; A = O[q]^40; A = (QP[q + A]/QP[q^13 + A])^2; s = A + 13*(q^2/A); CoefficientList[s, q] (* _Jean-François Alcover_, Nov 13 2015, adapted from PARI *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x + A) / eta(x^13 + A))^2; polcoeff( A + 13 * x^2 / A, n))}; /* _Michael Somos_, Jul 05 2012 */

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

%Y Cf. A034319, A121597, A133099.

%K sign,easy

%O -1,2

%A _N. J. A. Sloane_

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)