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A080054
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Expansion of f(q) / f(-q) in powers of q where f() is a Ramanujan theta function.
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9
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1, 2, 2, 4, 6, 8, 12, 16, 22, 30, 40, 52, 68, 88, 112, 144, 182, 228, 286, 356, 440, 544, 668, 816, 996, 1210, 1464, 1768, 2128, 2552, 3056, 3648, 4342, 5160, 6116, 7232, 8538, 10056, 11820, 13872, 16248, 18996, 22176, 25844, 30068, 34936, 40528
(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).
G.f. for pairs of partitions of type R.
G.f. for the number of partitions of 2n in which all odd parts occur with multplicity 2 and the even parts occur with multiplicity 1. Also g.f. for the number of partitions of 2n free of multiples of 4. All odd parts occur with even multiplicities. The even parts occur with multiplicity 1. - Noureddine Chair (n.chair(AT)rocketmail.com), Feb 10 2005
This is also the number of overpartitions of an integer into odd parts. - James A. Sellers (sellersj(AT)math.psu.edu), Feb 18 2008
The Higher Algebra reference on page 517 has an unnumbered example between 251 and 252: "If u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0, prove that (u^2-v^2)^6=16u^2v^2(1-u^8)(1-v^8). [PEMB. COLL. CAMB.]". It turns out that this is two forms of the modular equation of degree 5. - Michael Somos, May 12 2011
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REFERENCES
| B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054.
C. Bessenrodt, On pairs of partitions with steadily decreasing parts, J. Combin. Theory, A 99 (2002), 162-174. MR1911463 (2003c:11133)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London, 164 (1874), pp. 397-456, see pages 424 and 430.
A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed., G. Bell and Sons, 1895, p. 245, Art. 333.
H. S. Hall and S. R. Knight, Higher Algebra, Macmillan, 1957, p. 517.
Hirschhorn, M. D. and Sellers, J. A., Arithmetic Properties of Overpartitions into Odd Parts, Annals of Combinatorics 10, no. 3 (2006), 353-367
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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 - k^2)^(-1/8) = k'^(-1/4) in powers of the nome q = exp(-pi K'/K).
Expansion of eta(q^2)^3 / (eta(q^4) * eta(q)^2) in powers of q.
Euler transform of period 4 sequence [2, -1, 2, 0, ...].
(theta_3(q) / theta_4(q))^(1/2) = (phi(q) / phi(-q))^(1/2) = chi(q) / chi(-q) = psi(q) / psi(-q) = f(q) / f(-q) where phi, chi, psi, f are Ramanujan theta functions.
G.f.: A(x) = exp( 2*sum_{n>=0} sigma(2*n+1)/(2*n+1)*x^(2*n+1) ). - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 01 2004
G.f. satisfies: A(-x) = 1/A(x), (A(x)+A(-x))/2 = A(x^2)*A(x^4)^2, A(x) = sqrt((A(x^2)^4+1)/2) + sqrt((A(x^2)^4-1)/2). - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 27 2004
Another g.f.: 1/product_{ k>= 1 } (1+x^(2*k))*(1-x^(2*k-1))^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 29 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v^3) * (v + 2*u^3) - u * (u^3 - v). - Michael Somos, Aug 03 2011
G.f. A(x) satisfies 0 = f(A(x), A(x^5)) where f(u, v) = (u^2 - v^2)^6 - 16 * u^2 * v^2 * (1 - u^8) * (1 - v^8). - Michael Somos, May 12 2011
G.f. A(x) satisfies 0 = f(A(x), A(x^7)) where f(u, v) = (1 - u^8) * (1 - v^8) - (1 - u*v)^8. - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1/(32*t)) = 2^(-1/2)*g(t) where q = exp(2*Pi*i*t) and g() is g.f. for A029383. - Michael Somos, Aug 03 2011
G.f.: (theta_3/theta_4)^(1/2) = ((Sum_{k} x^(k^2))/(Sum_{k} (-x)^(k^2)))^(1/2) = Product_{k>0} (1 - x^(4k-2))/((1 - x^(4k-1))(1 - x^(4k-3)))^2.
G.f.: Product_{ k >= 1 } (1 + x^(2*k-1))*(1 + x^k) = product_{ k >= 1 } (1 + x^(2*k-1))/(1 - x^(2*k-1)).
A080015(n) = a(n) * (-1)^(n\2). A108494(n) = (-1)^n * a(n). Convolution inverse is A108494. Convolution square is A007096.
Empirical : sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(1/8). Simon Plouffe, Feb. 20, 2011.
Empirical : sum(exp(-Pi)^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = 2^(7/8)/2. Simon Plouffe, Feb. 20, 2011.
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EXAMPLE
| 1 + 2*q + 2*q^2 + 4*q^3 + 6*q^4 + 8*q^5 + 12*q^6 + 16*q^7 + 22*q^8 + 30*q^9 + ...
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MATHEMATICA
| a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m )^(-1/8), {q, 0, n}]] (* Michael Somos, Aug 03 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^(1/2) , {q, 0, n}] (* Michael Somos, Aug 03 2011 *)
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PROG
| (PARI) {a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + 2*x + O(x^2); while( m<n, m*=2; A = subst(A, x, x^2); A = sqrt((A^4 + 1) / 2) + 2 * sqrt((A^4 - 1) / 8)); polcoeff(A, n))}
(PARI) a(n)=polcoeff(exp(2*sum(k=0, n\2, sigma(2*k+1)/(2*k+1)*x^(2*k+1))), n) /*from Paul Hanna*/
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)), n))} /* Michael Somos, Jul 07 2005 */
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CROSSREFS
| Cf. A007096, A029838, A080015, A103258, A108494.
Sequence in context: A051466 A080015 * A108494 A078578 A018129 A091915
Adjacent sequences: A080051 A080052 A080053 * A080055 A080056 A080057
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KEYWORD
| nonn,easy
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AUTHOR
| Michael Somos, Jan 26 2003
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