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A058613
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McKay-Thompson series of class 30B for the Monster group with a(0) = 0.
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1
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1, 0, 4, 2, 6, 10, 15, 18, 37, 30, 57, 70, 105, 114, 178, 192, 285, 346, 465, 522, 751, 830, 1125, 1328, 1708, 1974, 2600, 2964, 3795, 4424, 5541, 6390, 8090, 9230, 11424, 13308, 16225, 18714, 22941, 26216, 31794, 36730, 44020, 50544, 60671, 69360, 82560, 94952
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OFFSET
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-1,3
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LINKS
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FORMULA
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G.f. T30B = 3 + e30A + 1 / e30A = 1 + e30C + 4 / e30C = -2 + e30D + 1 / e30D = -1 + e30F + 1 / e30F where e30A is g.f. A205826, e30C is g.f. A132321, e30D is g.f. A205962, and e30F is g.f. A205977.
a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(3/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
Expansion of A + 3 + 1/A, where A := (eta(q)*eta(q^6)*eta(q^10)*eta(q^15] )/( eta(q^2)*eta(q^3)*eta(q^5)*eta(q^30)))^3, in powers of q. - G. C. Greubel, Jun 22 2018
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EXAMPLE
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T30B = 1/q + 4*q + 2*q^2 + 6*q^3 + 10*q^4 + 15*q^5 + 18*q^6 + 37*q^7 + ...
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MATHEMATICA
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nmax = 50; QP = QPochhammer; A = x*O[x]^(nmax + 1); A = (QP[A + x^3]*QP[A + x^5]*QP[A + x^6]*QP[A + x^10])/(QP[A + x]*QP[A + x^2]*QP[A + x^15]*QP[A + x^30]); a[n_] := SeriesCoefficient[x^2/A + A - x, n + 1]; Table[a[n], {n, -1, nmax}] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
eta[q_]:= q^(1/24)*QPochhammer[q]; A := (eta[q]*eta[q^6]*eta[q^10]* eta[q^15]/(eta[q^2]*eta[q^3]*eta[q^5]*eta[q^30]))^3; a:=CoefficientList[ Series[q*(A + 3 + 1/A), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 22 2018 *)
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PROG
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(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = eta(x^3 + A) * eta(x^5 + A) * eta(x^6 + A) * eta(x^10 + A) / (eta(x + A) * eta(x^2 + A) * eta(x^15 + A) * eta(x^30 + A)); polcoeff( -x + A + x^2 / A, n))} /* Michael Somos, Feb 02 2012 */
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CROSSREFS
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Cf. A000521, A007240, A014708, A007241, A007267, A045478, A058732, A132321, A205826, A205962, A205977.
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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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