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A187145
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McKay-Thompson series of class 12I for the Monster group with a(0) = 3.
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0
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1, 3, 2, 0, 1, 0, 0, 0, -2, 0, -2, 0, 2, 0, 4, 0, 3, 0, -4, 0, -8, 0, -4, 0, 5, 0, 14, 0, 7, 0, -8, 0, -20, 0, -12, 0, 14, 0, 28, 0, 17, 0, -20, 0, -44, 0, -24, 0, 28, 0, 66, 0, 36, 0, -40, 0, -90, 0, -52, 0, 56, 0, 124, 0, 71, 0, -80, 0, -176, 0, -96, 0, 109, 0, 244, 0, 133, 0, -144
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OFFSET
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-1,2
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COMMENTS
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Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0. .oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
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D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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LINKS
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Table of n, a(n) for n=-1..77.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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Expansion of (1/q) * (phi(q) * psi(q)) / (psi(q^3) * psi(q^6)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^7 * eta(q^3) / (eta(q)^3 * eta(q^4)^2 * eta(q^6) * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ 3, -4, 2, -2, 3, -4, 3, -2, 2, -4, 3, 0, ...].
a(2*n) = 0 unless n=0. a(2*n - 1) = A058487(n).
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EXAMPLE
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1/q + 3 + 2*q + q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 - 4*q^17 + ...
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PROG
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(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^12 + A)^2), n))}
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CROSSREFS
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Cf. A187144, A187143, A187130, A058487.
Sequence in context: A133209 A144553 A187130 * A131290 A116604 A138741
Adjacent sequences: A187142 A187143 A187144 * A187146 A187147 A187148
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Mar 05 2011
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STATUS
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approved
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