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A080054
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G.f.: Product_{n >= 0} (1+x^(2n+1))/(1-x^(2n+1)).
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40
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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
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OFFSET
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0,2
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COMMENTS
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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 multiplicity 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, Feb 10 2005
This is also the number of overpartitions of an integer into odd parts. - James A. Sellers, 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
Let F(x) = Product_{n >= 0} (1 + x^(2*n+1))/(1 - x^(2*n+1)). The simple continued fractions expansions of the real numbers F(1/n) may be predictable - the partial denominators may be polynomial or quasi-polynomial in n. An example is given below. - Peter Bala, Nov 03 2019
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REFERENCES
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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.
A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed., G. Bell and Sons, 1895, p. 245, Art. 333.
J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. VI
J. W. L. Glaisher, On Some Continued Fractions, Messenger of Mathematics, 7 (1878), pp. 67-68, see p. 68
H. S. Hall and S. R. Knight, Higher Algebra, Macmillan, 1957, p. 517.
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LINKS
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FORMULA
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Expansion of f(q) / f(-q) in powers of q where f() is a Ramanujan theta function.
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, 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, Mar 27 2004
Another g.f.: 1/product_{ k>= 1 } (1+x^(2*k))*(1-x^(2*k-1))^2. - Vladeta Jovovic, 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 the g.f. for A029838. - Michael Somos, Aug 03 2011
G.f.: (theta_3/theta_4)^(1/2) = ((Sum_{k in Z} x^(k^2))/(Sum_{k in Z} (-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)).
G.f.: 1 + 2*x / (1 - x) + 2*x^3 * (1 + x) / ((1 - x)*(1 - x^2)) + 2*x^6 * (1 + x)*(1 + x^2) / ((1 - x)*(1 - x^2)*(1 - x^3)) + ... [Glaisher 1876] - Michael Somos, Jun 20 2012
G.f.: 1 / (1 - 2*x / (1 + x - (x^2 - x^4) / (1 + x^3 - (x^3 - x^7) / (1 + x^5 - (x^4 - x^10) / (1 + x^7 - ...))))) [Glaisher 1878] - Michael Somos, Jun 24 2012
Empirical : Sum_{n>=0} exp(-Pi)^n * a(n) = 2^(1/8). - Simon Plouffe, Feb 20 2011
Empirical : Sum_{n>=0} (-exp(-Pi))^n * a(n) = 1/2^(1/8). - Simon Plouffe, Feb 20 2011
a(n) ~ Pi * BesselI(1, Pi*sqrt(n/2)) / (4*sqrt(n)) ~ exp(Pi*sqrt(n/2)) / (2^(9/4) * n^(3/4)) * (1 - 3/(4*Pi*(sqrt(2*n))) - 15/(64*Pi^2*n)). - Vaclav Kotesovec, Aug 23 2015, extended Jan 09 2017
Simon Plouffe's empirical observations are true. Furthermore, for every positive rational p, Sum_{n>=0} exp(-Pi*sqrt(p))^n * a(n) = 1/(Sum_{n>=0} (-exp(-Pi*sqrt(p)))^n * a(n)) is an algebraic number (see the MathOverflow link). - Vladimir Reshetnikov, Nov 23 2016
G.f.: f(x,x^3)/f(-x,-x^3) = ( Sum_{n = -oo..oo} x^(n*(2*n-1) )/(
Sum_{n = -oo..oo} (-1)^n*x^(n*(2*n-1) ), where f(a,b) = Sum_{n = -oo..oo} a^(n*(n+1)/2)*b^(n*(n-1)/2) is Ramanujan's 2-variable theta function. - Peter Bala, Feb 05 2021
G.f. A(q) = (-lambda(-q)/lambda(q))^(1/8), where lambda(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... is the elliptic modular function in powers of the nome q = exp(i*Pi*t), the g.f. of A115977; lambda(q) = k(q)^2, where k(q) = (theta_2(q) / theta_3(q))^2 is the elliptic modulus. - Peter Bala, Sep 26 2023
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EXAMPLE
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G.f. = 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 + ...
F(x) := Product_{n >= 0} (1 + x^(2*n+1))/(1 - x^(2*n+1)).
Simple continued fraction expansions of F(1/(2*n)):
n=2 [1; 1, 2, 1, 1, 1, 1, 2, 1, 2, 33, 1, 3, 7, 4, 33, 1, 8, 4, 2, 1,...]
n=3 [1; 2, 2, 2, 1, 1, 2, 2, 2, 2, 110, 1, 2, 46, 3, 110, 1, 3, 12, 1, 7,...]
n=4 [1; 3, 2, 3, 1, 1, 3, 2, 3, 2, 259, 1, 1, 1, 2, 15, 2, 1, 2, 259, 1,...]
n=5 [1; 4, 2, 4, 1, 1, 4, 2, 4, 2, 504, 1, 1, 1, 1, 78, 1, 1, 2, 504, 1,...]
n=6 [1; 5, 2, 5, 1, 1, 5, 2, 5, 2, 869, 1, 1, 2, 2, 23, 2, 2, 2, 869, 1,...]
n=7 [1; 6, 2, 6, 1, 1, 6, 2, 6, 2, 1378, 1, 1, 2, 1, 110, 1, 2, 2, 1378, 1,...]
n=8 [1; 7, 2, 7, 1, 1, 7, 2, 7, 2, 2055, 1, 1, 3, 2, 31, 2, 3, 2, 2055, 1,...]
n=9 [1; 8, 2, 8, 1, 1, 8, 2, 8, 2, 2924, 1, 1, 3, 1, 142, 1, 3, 2, 2924, 1,...]
The sequence of 10th denominators [33,110,259,504,...] appears to be given by the polynomial 4*n^3 + n - 1.
The sequence of 15th denominators [15,78,23,110,31,142,...], starting at n = 4, appears to be quasi-polynomial in n, with constituent polynomials 4*n - 1 and 16*n - 2.
(End)
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-2) +add(2*b(n-i*j, i-2), j=1..n/i)))
end:
a:= n-> b(n, n-1+irem(n, 2)):
# alternative program using expansion of f(x, x^3) / f(-x, -x^3):
with(gfun): series( add(x^(n*(2*n-1)), n = -8..8)/add((-1)^n*x^(n*(2*n-1)), n = -8..8), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021
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MATHEMATICA
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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 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q] / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-1}, {}, q^2, q], {q, 0, n}]; (* Michael Somos, May 10 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 2] + Sum[2*b[n - i*j, i - 2], {j, 1, n/i}]]];
a[n_] := b[n, n - 1 + Mod[n, 2]];
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PROG
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(PARI) {a(n) = my(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) /* Paul D. Hanna */
(PARI) {a(n) = my(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
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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