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A132322
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McKay-Thompson series of class 46A for the Monster group with a(0) = -1.
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4
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1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -10, 12, -12, 13, -15, 17, -18, 19, -22, 25, -27, 28, -32, 36, -38, 41, -46, 51, -54, 58, -64, 71, -76, 81, -89, 99, -105, 112, -123, 134, -143, 153, -167, 182, -194, 207, -225, 244, -260, 277, -301, 325, -346, 369, -398, 429, -458
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OFFSET
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-1,9
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^-1 * chi(-q) * chi(-q^23) in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q) * eta(q^23) / (eta(q^2) * eta(q^46)) in powers of q.
Euler transform of period 46 sequence [ -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 2 * u + 2 * u * v.
G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A092833.
G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(23*k)))^-1.
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018
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EXAMPLE
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G.f. = 1/q - 1 - q^2 + q^3 - q^4 + q^5 - q^6 + 2*q^7 - 2*q^8 + 2*q^9 - 2*q^10 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q^23, q^46] / q, {q, 0, n}]; (* Michael Somos, Mar 23 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 + A) * eta(x^23 + A) / (eta(x^2 + A) * eta(x^46 + 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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