0,2

Expansion of f(q)/f(-q) where f() is a Ramanujan theta function.

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

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)

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

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's 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.yu), Mar 29 2004

G.f. A(x) satisfies 0=f(A(x), A(x^7)) where f(u, v)=(1-u^8)(1-v^8)-(1-uv)^8 . - Michael Somos Jan 01 2006

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.

(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 */

Cf. A007096, A103258. A080015(n)=a(n)*(-1)^(n\2).

Sequence in context: A116859 A051466 A080015 this_sequence A108494 A078578 A018129

Adjacent sequences: A080051 A080052 A080053 this_sequence A080055 A080056 A080057

nonn,easy

Michael Somos, Jan 26 2003