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McKay-Thompson series of class 24B for the Monster group with a(0) = 2.
2

%I #15 Jan 28 2018 22:54:03

%S 1,2,3,8,11,16,31,40,58,96,125,176,262,336,457,640,819,1088,1464,1864,

%T 2420,3168,3991,5088,6533,8160,10267,12976,16061,19968,24912,30576,

%U 37648,46464,56616,69136,84518,102288,123961,150304,180805,217664,262042,313472

%N McKay-Thompson series of class 24B for the Monster group with a(0) = 2.

%H G. C. Greubel, <a href="/A212771/b212771.txt">Table of n, a(n) for n = -1..1000</a>

%F Expansion of (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12) / (eta(q) * eta(q^3) * eta(q^8) * eta(q^24)) )^2 in powers of q.

%F Euler transform of period 24 sequence [ 2, 0, 4, -2, 2, 0, 2, 0, 4, 0, 2, -4, 2, 0, 4, 0, 2, 0, 2, -2, 4, 0, 2, 0, ...].

%F Expansion of b(q^2) * c(q^2) * b(q^4) * c(q^4) / (b(q) * c(q) * b(q^8) * c(q^8)) in powers of q where b(), c() are cubic AGM theta functions.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).

%F a(n) = A058572(n) unless n=0. a(2*n) = 2 * A123861(n). Convolution inverse of A212770.

%F a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(5/4)*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%e 1/q + 2 + 3*q + 8*q^2 + 11*q^3 + 16*q^4 + 31*q^5 + 40*q^6 + 58*q^7 + ...

%t QP = QPochhammer; s = (QP[q^2]*QP[q^4]*QP[q^6]*(QP[q^12]/(QP[q]*QP[q^3]* QP[q^8]*QP[q^24])))^2 + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n = n+1; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^24 + A)) )^2, n))}

%Y Cf. A058572, A123861, A212770.

%K nonn

%O -1,2

%A _Michael Somos_, May 26 2012