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A022597
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Expansion of Product_{m >= 1} (1 + q^m)^(-2).
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13
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1, -2, 1, -2, 4, -4, 5, -6, 9, -12, 13, -16, 21, -26, 29, -36, 46, -54, 62, -74, 90, -106, 122, -142, 171, -200, 227, -264, 311, -358, 408, -470, 545, -626, 709, -810, 933, -1062, 1198, -1362, 1555, -1760, 1980, -2238, 2536, -2858, 3205, -3602, 4063, -4560, 5092, -5704, 6400, -7150, 7966
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OFFSET
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0,2
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COMMENTS
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McKay-Thompson series of class 24J for the Monster group.
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REFERENCES
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T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2.
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LINKS
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FORMULA
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Expansion of q^(1/12) * (eta(q) / eta(q^2))^2 in powers of q.
Euler transform of period 2 sequence [ -2, 0, ...]. - Michael Somos, Sep 10 2005
Expansion of chi(-x)^2 in powers of x where chi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A022567.
G.f.: Product_{k>0} (1 + x^k)^-2.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
G.f.: exp(-2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
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EXAMPLE
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G.f. = 1 - 2*x + x^2 - 2*x^3 + 4*x^4 - 4*x^5 + 5*x^6 - 6*x^7 + 9*x^8 + ...
T24J = 1/q - 2*q^11 + q^23 - 2*q^35 + 4*q^47 - 4*q^59 + 5*q^71 - 6*q^83 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2, {x, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, n}]^-2, {x, 0, n}]; (* Michael Somos, Jul 11 2011 *)
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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 + A) / eta(x^2 + A))^2, n))}; /* Michael Somos, Sep 10 2005 */
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CROSSREFS
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Cf. A089814 (expansion of Product_{k>=1}(1-q^(10k-5))^2).
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KEYWORD
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sign,nice,easy
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AUTHOR
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STATUS
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approved
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