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A080054 G.f.: Product_{n >= 0} (1+x^(2n+1))/(1-x^(2n+1)). 37
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; text; internal format)
OFFSET

0,2

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 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

Convolution of A000009 and A000700. - Vaclav Kotesovec, Aug 23 2015

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

REFERENCES

Ardonne, Eddy, Rinat Kedem, and Michael Stone. "Filling the Bose sea: symmetric quantum Hall edge states and affine characters." Journal of Physics A: Mathematical and General 38.3 (2005): 617. - From N. J. A. Sloane, Apr 24 2014

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.

Chen, Shi-Chao. On the number of overpartitions into odd parts. Discrete Math. 325 (2014), 32--37. MR3181230. But beware of typos in the g.f. on page 32. - N. J. A. Sloane, Apr 24 2014

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.

Hirschhorn, M. D. and Sellers, J. A., Arithmetic Properties of Overpartitions into Odd Parts, Annals of Combinatorics 10, no. 3 (2006), 353-367

M. Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.

Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See q-bar(n).

Vladimir Reshetnikov, A conjecture about algebraic values of (-q;-q)_oo/(q;q)_oo, Math Overflow Posting, Nov 23 2016, with proof supplied by Noam D. Elkies.

Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

a(n) = (-1)^floor(n/2) * A080015(n) = (-1)^n * A108494(n). Convolution inverse is A108494. Convolution square is A007096.

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

EXAMPLE

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 + ...

From Peter Bala, Nov 03 2019: (Start)

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)

MAPLE

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)):

seq(a(n), n=0..50);  # Alois P. Heinz, Feb 10 2014

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 *)

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]];

Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Nov 05 2017, after Alois P. Heinz *)

PROG

(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 */

CROSSREFS

Cf. A007096, A029838, A080015, A103258, A108494, A261610, A261611.

Sequence in context: A261156 A080015 A210030 * A108494 A078578 A323446

Adjacent sequences:  A080051 A080052 A080053 * A080055 A080056 A080057

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Jan 26 2003

EXTENSIONS

Definition simplified by N. J. A. Sloane, Apr 24 2014

STATUS

approved

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Last modified December 11 07:18 EST 2019. Contains 329914 sequences. (Running on oeis4.)