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A124863
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Expansion of 1 / chi(q)^12 in powers of q where chi() is a Ramanujan theta function.
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3
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1, -12, 78, -376, 1509, -5316, 16966, -50088, 138738, -364284, 913824, -2203368, 5130999, -11585208, 25444278, -54504160, 114133296, -234091152, 471062830, -931388232, 1811754522, -3471186596, 6556994502, -12222818640, 22502406793
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of q^(-1/2) * (k * k') / 4 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.
Expansion of q^(-1/2) * (eta(q) * eta(q^4) / eta(q^2)^2)^12 in powers of q.
Euler transform of period 4 sequence [ -12, 12, -12, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 22 2011
G.f.: Product_{k>0} (1 + (-x)^k)^12 = Product_{k>0} 1/(1 + x^(2*k - 1))^12. [corrected by Vaclav Kotesovec, Nov 16 2017]
G.f.: T(0), where T(k) = 1 - 1/(1 - 1/(1 - 1/(1+(-x)^(k+1))^12/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)) / (128*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
G.f.: exp(-12*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018
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EXAMPLE
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G.f. = 1 - 12*x + 78*x^2 - 376*x^3 + 1509*x^4 - 5316*x^5 + 16966*x^6 - 50088*x^7 + ...
G.f. = q - 12*q^3 + 78*q^5 - 376*q^7 + 1509*q^9 - 5316*q^11 + 16966*q^13 - 50088*q^15 + ...
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MATHEMATICA
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a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m/16/q)^(-1/2), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := SeriesCoefficient[1/Product[1 + q^k, {k, 1, n, 2}]^12, {q, 0, n}]; (* Michael Somos, Jul 22 2011, fixed by Vaclav Kotesovec, Nov 16 2017 *)
nmax = 30; CoefficientList[Series[Product[1/(1 + x^(2*k - 1))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)
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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^4 + A) / eta(x^2 + A)^2)^12, 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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