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McKay-Thompson series of class 19A for the Monster group with a(0) = 3.
3

%I #14 Sep 07 2017 08:30:37

%S 1,3,6,10,21,36,61,96,156,232,357,522,768,1092,1563,2174,3039,4164,

%T 5695,7686,10362,13792,18333,24138,31706,41316,53712,69348,89319,

%U 114396,146114,185724,235482,297252,374316,469578,587646,732888,911961,1131250

%N McKay-Thompson series of class 19A for the Monster group with a(0) = 3.

%H Vaclav Kotesovec, <a href="/A136569/b136569.txt">Table of n, a(n) for n = -1..10000</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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F G.f.: x^(-1) * ( G(x) * G(x^19) + x^4 * H(x) * H(x^19) )^3 where G() is g.f. of A003114 and H() is g.f. of A003106.

%F a(n) ~ exp(4*Pi*sqrt(n/19)) / (sqrt(2)*19^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%e 1/q + 3 + 6*q + 10*q^2 + 21*q^3 + 36*q^4 + 61*q^5 + 96*q^6 + 156*q^7 + ...

%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])^3 + O[x]^40; CoefficientList[s, x] (* _Jean-François Alcover_, Nov 15 2015 *)

%o (PARI) {a(n) = local(A, A1, A2); if( n<-1, 0, n = 2*n + 2 ; 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); A = ( x^4 / A1 / A2 - (A1 + A2) / 4 / x )^3; polcoeff( A, n ))}

%Y Cf. A058549(n) = a(n) unless n=0.

%K nonn

%O -1,2

%A _Michael Somos_, Jan 07 2008