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McKay-Thompson series of class 60D for the Monster group with a(0) = -1.
3

%I #29 Sep 08 2022 08:45:37

%S 1,-1,-1,1,0,0,0,-1,1,1,-1,-1,1,0,-1,2,0,-2,2,-1,0,2,-4,0,5,-1,-4,2,1,

%T -2,3,-3,-2,7,-5,-2,8,-6,-5,8,1,-5,2,-2,-1,12,-11,-10,21,-6,-10,13,-7,

%U -4,11,-7,-4,14,-13,-10,33,-14,-28,32,-3,-12,18,-24,1,36,-27,-22,44,-13,-35,50,-13,-36,46,-26,-6,56,-63,-22,89,-30

%N McKay-Thompson series of class 60D for the Monster group with a(0) = -1.

%H Seiichi Manyama, <a href="/A143751/b143751.txt">Table of n, a(n) for n = -1..10000</a>

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/retaprod.html">A Remarkable eta-product Identity </a>

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

%F Expansion of eta(q) * eta(q^12) * eta(q^15) * eta(q^20) / (eta(q^3) * eta(q^4) * eta(q^5) * eta(q^60)) in powers of q.

%F Expansion of F(q) * F(q^2) in powers of q^3 where F(q) is the g.f. of A112215.

%F Euler transform of a period 60 sequence.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A143752.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 + v^2) * (1 + u + v) * (u + v + u*v) - u*v * (1 + 2*u + 2*v + u*v)^2.

%F G.f.: (x * Product_{k>0} P(30, x^k) * P(60, x^k))^(-1) where P(n, x) is the n-th cyclotomic polynomial.

%F A058728(n) = a(n) unless n=0. Convolution inverse of A143752.

%e G.f. = 1/q - 1 - q + q^2 - q^6 + q^7 + q^8 - q^9 - q^10 + q^11 - q^13 + 2*q^14 + ...

%t QP = QPochhammer; s = QP[q]*QP[q^12]*QP[q^15]*(QP[q^20]/(QP[q^3]*QP[q^4]* QP[q^5]*QP[q^60])) + O[q]^90; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015, adapted from PARI *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A) / (eta(x^3 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^60 + A)), n))};

%o (Magma) S<x> := PowerSeriesRing(RationalField());Coefficients( DedekindEta(x)*DedekindEta(x^12)*DedekindEta(x^15)*DedekindEta(x^20)/( DedekindEta(x^3) *DedekindEta(x^4)*DedekindEta(x^5)*DedekindEta(x^60))); // _G. C. Greubel_, Mar 04 2018

%Y Cf. A058728, A112215, A143752.

%K sign

%O -1,16

%A _Michael Somos_, Aug 31 2008