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A138688
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McKay-Thompson series of class 24I for the Monster group with a(0) = 2.
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2
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1, 2, 4, 6, 11, 18, 28, 42, 62, 90, 128, 180, 250, 342, 464, 624, 831, 1098, 1440, 1878, 2432, 3132, 4012, 5112, 6485, 8190, 10300, 12900, 16097, 20016, 24804, 30636, 37724, 46314, 56700, 69228, 84302, 102402, 124088, 150024, 180973, 217836, 261664, 313680
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OFFSET
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-1,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of psi(q^4) * phi(-q^3) / (phi(-q) * psi(q^12)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^3)^2 * eta(q^8)^2 * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6) * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ 2, 1, 0, 2, 2, 0, 2, 0, 0, 1, 2, 0, 2, 1, 0, 0, 2, 0, 2, 2, 0, 1, 2, 0, ...].
G.f.: (G(x) * G(x^24) + x^5 * H(x) * H(x^24))^2 * (G(x^4) * G(x^6) + x^2 * H(x^4) * H(x^6)) where G() and H() are Rogers-Ramanujan functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).
a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(5/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
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EXAMPLE
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G.f. = 1/q + 2 + 4*q + 6*q^2 + 11*q^3 + 18*q^4 + 28*q^5 + 42*q^6 + 62*q^7 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^2] EllipticTheta[ 4, 0, q^3] / (EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^6]), {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(3*k))^2 * (1-x^(4*k)) * (1+x^(4*k))^2 * (1+x^(6*k)) / ((1-x^k) * (1-x^(24*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
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PROG
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(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^8 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)^2), 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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