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A104794
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Expansion of theta_4(q)^2 in powers of q.
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3
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1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, 0, 0, -12, 8, 0, 0, -8, 0, 0, 4, 0, 8, 0, 4, -8, 0, 0, 8, -8, 0, 0, 0, -8, 0, 0, 0, -4, 12, 0, 8, -8, 0, 0, 0, 0, 8, 0, 0, -8, 0, 0, 4, -16, 0, 0, 8, 0, 0, 0, 4, -8, 8, 0, 0, 0, 0, 0, 8, -4, 8, 0, 0, -16, 0, 0, 0, -8, 8, 0, 0, 0, 0, 0, 0, -8, 4, 0, 12
(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).
In the Arithmetic-Geometric Mean, if a = theta_3(q)^2, b = theta_4(q)^2 then a' := (a+b)/2 = theta_3(q^2)^2, b' := sqrt(a*b) = theta_4(q^2)^2.
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.
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LINKS
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
| Expansion of phi(-q)^2 = 2 * phi(q^2)^2 - phi(q)^2 = (phi(q) - 2*phi(q^4))^2 = f(-q)^3 / psi(q) = phi(-q^2)^4 / phi(q)^2 = psi(-q)^4 / psi(q^2)^2 = psi(q)^2 * chi(-q)^6 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (1-k^2)^(1/2) K(k^2) / (pi/2) in powers of q where q is Jacobi's nome, k is the ellptic modulus and K() is the complete elliptic integral.
Expansion of eta(q)^4 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -4, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u^2 + v^2) - 2*u*w^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 - 2*u1*u3 + 4*u2*u6 - 3*u3^2.
Moebius transform is period 8 sequence [ -4, 8, 4, 0, -4, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 16 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A008441.
G.f.: theta_4(q)^2 = (Sum_{k} (-q)^(k^2))^2 = (Product_{k>0} (1 - q^(2*k)) * (1 - q^(2*k - 1))^2)^2.
a(4*n + 3) = 0. A004018(n) = (-1)^n * a(n) = a(2*n). -4 * A008441(n) = a(4*n+1). -4 * A113652(n) = a(n) unless n=0. 4 * A122865(n) = a(6*n+2). 4 * A122856(n) = a(6*n+4). -4 * A113407(n) = a(8*n+1). -8 * A053692(n) = a(8*n+5).
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EXAMPLE
| 1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...
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MATHEMATICA
| a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] ^2 , {q, 0, n}]
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[1 - m] EllipticK[m] / (Pi/2), {q, 0, n}]]
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/4) EllipticK[m] / (Pi/2), {q, 0, 2 n}]]
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PROG
| (PARI) {a(n) = if( n<1, n==0, (-1)^n * 4 * sumdiv(n, d, (d%4==1) - (d%4==3)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 / eta(x^2 + A)^2, n ))}
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 4 * sum( k=1, n, (-x)^k / (1 + x^(2*k)), x * O(x^n)), n))}
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CROSSREFS
| Cf. A004018, A008441, A053692, A113407, A113652, A122856, A122865.
Sequence in context: A138518 A155836 A164613 * A004018 A028658 A169784
Adjacent sequences: A104791 A104792 A104793 * A104795 A104796 A104797
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Mar 26 2005
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