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A124863
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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.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of 1 / chi(q)^12 in powers of q where chi() is a Ramanujan theta function.
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 + x^(2*k - 1))^12.
a(n) = (-1)^n * A022577(n). Convolution inverse of A112142. Convolution square is A100130.
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EXAMPLE
| 1 - 12*x + 78*x^2 - 376*x^3 + 1509*x^4 - 5316*x^5 + 16966*x^6 - 50088*x^7 + ...
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
| 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[ Product[1 + q^k, {k, 1, n, 2}]^12, {q, 0, n}] (* Michael Somos, Jul 22 2011 *)
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PROG
| (PARI) {a(n) = local(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
| Cf. A022577, A100130, A112142.
Sequence in context: A008494 A001288 A121665 * A022577 A189493 A199492
Adjacent sequences: A124860 A124861 A124862 * A124864 A124865 A124866
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Nov 10 2006
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