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A136569
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McKay-Thompson series of class 19A for the Monster group with a(0) = 3.
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3
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1, 3, 6, 10, 21, 36, 61, 96, 156, 232, 357, 522, 768, 1092, 1563, 2174, 3039, 4164, 5695, 7686, 10362, 13792, 18333, 24138, 31706, 41316, 53712, 69348, 89319, 114396, 146114, 185724, 235482, 297252, 374316, 469578, 587646, 732888, 911961, 1131250
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OFFSET
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-1,2
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LINKS
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FORMULA
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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.
a(n) ~ exp(4*Pi*sqrt(n/19)) / (sqrt(2)*19^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
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EXAMPLE
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1/q + 3 + 6*q + 10*q^2 + 21*q^3 + 36*q^4 + 61*q^5 + 96*q^6 + 156*q^7 + ...
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MATHEMATICA
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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 *)
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PROG
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(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 ))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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