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A132130
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McKay-Thompson series of class 10D for the Monster group with a(0) = 6.
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4
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1, 6, 21, 62, 162, 378, 819, 1680, 3276, 6138, 11145, 19662, 33840, 57048, 94362, 153432, 245757, 388218, 605466, 933414, 1423614, 2149586, 3215844, 4769544, 7016572, 10243896, 14848809, 21378276, 30582360, 43484304, 61473438, 86428896
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OFFSET
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-1,2
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COMMENTS
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The g.f. is denoted by x_10 in Cooper 2012.
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LINKS
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FORMULA
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Expansion of q^(-1) * (chi(-q^5) / chi(-q))^6 in powers of q where chi() is a Ramanujan theta function.
Expansion of (eta(q^2) * eta(q^5) / (eta(q) * eta(q^10)))^6 in powers of q.
Euler transform of period 10 sequence [ 6, 0, 6, 0, 0, 0, 6, 0, 6, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (v - u^2) * (v - w^2) - u*w * (12*(1 + v^2) - 20*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = f(t) where q = exp(2 Pi i t).
G.f.: x^(-1) * (Product_{k>0} (1 + x^k) / (1 + x^(5*k)))^6.
G.f.: 1 / ( x * Product_{k>0} P(10,x^k)^6 ) where P(n,x) is the n-th cyclotomic polynomial.
a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(3/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
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EXAMPLE
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G.f. = 1/q + 6 + 21*q + 62*q^2 + 162*q^3 + 378*q^4 + 819*q^5 + 1680*q^6 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q^-1 (QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2])^6, {q, 0, n}]; (* Michael Somos, Dec 07 2013 *)
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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^5 + A) / (eta(x + A) * eta(x^10 + A)))^6, 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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EXTENSIONS
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STATUS
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approved
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