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A112199 McKay-Thompson series of class 57A for the Monster group. 2

%I

%S 1,1,1,1,3,2,4,4,5,6,8,9,12,13,16,18,23,25,31,36,43,48,57,64,76,86,99,

%T 112,131,146,169,190,217,243,278,310,353,394,446,498,562,624,704,781,

%U 877,972,1088,1204,1345,1488,1656,1829,2033,2240,2486,2738,3030,3334

%N McKay-Thompson series of class 57A for the Monster group.

%H G. C. Greubel, <a href="/A112199/b112199.txt">Table of n, a(n) for n = 0..5000</a>

%H K. Bringmann and H. Swisher, <a href="http://dx.doi.org/10.1090/S0002-9939-07-08735-7">On a conjecture of Koike on identities between Thompson series and Roger-Ramanujan functions</a>, Proc. Amer. Math. Soc. 135 (2007), 2317-2326. See page 2325 (A.5)

%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. See page 336.

%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 G.f.: G(x) * G(x^19) + x^4 * H(x) * H(x^19) where G() is g.f. of A003114 and H() is g.f. of A003106.

%F a(n) ~ exp(4*Pi*sqrt(n/57)) / (sqrt(2)*57^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%e T57A = 1/q +q^2 +q^5 +q^8 +3*q^11 +2*q^14 +4*q^17 +4*q^20 +...

%t QP = QPochhammer; G[x_] := 1/(QP[x, x^5]*QP[x^4, x^5]); H[x_] := 1/(QP[x^2, x^5]*QP[x^3, x^5]); s = (G[x]*G[x^19] + x^4*H[x]*H[x^19]) + O[x]^60; CoefficientList[s, x] (* _Jean-Fran├žois Alcover_, Nov 15 2015 *)

%o (PARI) {a(n) = local(A, A1, A2); if( n<0, 0, n = 2*n ; A = x^3 * O(x^n) ; A1 = ( eta(x + A) * eta(x^19 + A) / eta(x^2 + A) / eta(x^38 + A) )^2; A2 = -subst(A1, x, -x); polcoeff( x^4 / A1 / A2 - (A1 + A2) / 4 / x, n))} \\ _Michael Somos_, Jan 07 2008

%Y Cf. A136569.

%K nonn

%O 0,5

%A _Michael Somos_, Aug 28 2005

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Last modified June 14 10:04 EDT 2021. Contains 345025 sequences. (Running on oeis4.)