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A112212
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McKay-Thompson series of class 84C for the Monster group.
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4
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1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 3, 3, 4, 4, 4, 6, 7, 7, 7, 9, 10, 12, 13, 14, 17, 18, 19, 22, 26, 28, 29, 34, 38, 41, 44, 50, 57, 60, 65, 72, 81, 86, 94, 105, 114, 124, 133, 146, 161, 174, 187, 204, 224, 240, 258, 282, 309, 332, 354, 386, 419, 450, 481, 524, 569, 606
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OFFSET
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0,8
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COMMENTS
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Given g.f. A(x), the first term of the left side of Cayley's identity is A(q). - Michael Somos, Dec 03 2013
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REFERENCES
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A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.
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LINKS
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FORMULA
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Expansion of q^(1/3) * eta(q^2)^2 * eta(q^14)^2 / (eta(q) * eta(q^4) * eta(q^7) * eta(q^28)) in powers of q. - Michael Somos, Dec 03 2013
Euler transform of period 28 sequence [1, -1, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 2, -1, 1, 0, 1, -1, 1, 0, ...]. - Michael Somos, Dec 03 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 03 2013
G.f.: Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(14*k - 7)). - Michael Somos, Dec 03 2013
a(n) ~ exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
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EXAMPLE
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G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 3*x^10 + ...
T84C = 1/q + q^2 + q^8 + q^11 + q^14 + q^17 + 2*q^20 + 3*q^23 + 2*q^26 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^7, x^14], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] Product[ 1 + x^k, {k, 7, n, 14}], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^14 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A)), n))}; /* Michael Somos, Dec 03 2013 */
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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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