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A225849
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McKay-Thompson series of class 20C for the Monster group with a(0) = 3.
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3
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1, 3, 1, -2, 2, 2, -1, 0, -4, 2, 5, -2, 0, -8, 2, 8, -3, 2, -14, 6, 14, -6, 4, -24, 12, 24, -11, 4, -40, 16, 38, -16, 5, -62, 24, 60, -24, 10, -94, 40, 91, -38, 18, -144, 62, 136, -57, 24, -214, 88, 201, -82, 30, -308, 122, 288, -117, 48, -440, 180, 410
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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 q^(-1) * f(q)^3 / (f(-q^2) * f(-q^5)* f(-q^20)) in powers of q where f() is a Ramanujan theta function.
Expansion of q^(-1) * f(q, q)^2 / (f(q, q^9) * f(q^3, q^7)) in powers of q where f(, ) is Ramanujan's general theta functions.
Expansion of q^(-1) * (phi(q) / phi(q^5))^2 * chi(q^5)^5 / chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2)^8 / ( eta(q)^3 * eta(q^4)^3 * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 3, -5, 3, -2, 4, -5, 3, -2, 3, -4, 3, -2, 3, -5, 4, -2, 3, -5, 3, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 4) * (u - 5) * v * (v - 4) * (v - 5).
G.f.: (1/x) * Product_{k>0} (1 - (-x)^k)^3 / ((1 - x^(2*k)) * (1 - x^(5*k)) * (1 - x^(20*k))).
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EXAMPLE
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G.f. = 1/q + 3 + q - 2*q^2 + 2*q^3 + 2*q^4 - q^5 - 4*q^7 + 2*q^8 + 5*q^9 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 4 + (1/q) QPochhammer[ q^5, q^10]^5 / QPochhammer[ q, q^2], {q, 0, n}];
a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ -q]^3 / (QPochhammer[ q^2] QPochhammer[ q^5] QPochhammer[ q^20]), {q, 0, n}];
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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)^8 / ( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^5 + A) * eta(x^20 + A)), n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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