0,2

Number of partitions of 2n into parts with 2 types c, c* of each part. The even parts appears with multiplicity 1 for each type. The odd parts appears with multiplicity 2 (cc or c*c* but not cc*, that is, no mixing is allowed). E.g. a(4)=8 because of 44*, 22*, 211, 21*1*, 2*1*1*, 2*11, 111*1*. - Noureddine Chair (n.chair(AT)rocketmail.com), Jan 27 2005

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

Self-convolution of A080054. Euler transform of period 4 sequence [4, -2, 4, 0, ...]. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 22 2005

Expansion of eta(q^2)^6/(et(q)^4eta(q^4)^2) in powers of q.

Euler transform of period 4 sequence [4, -2, 4, 0, ...]. - Michael Somos, Jul 07 2005

G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=1+u^2-2uv^2. - Michael Somos Jul 07 2005

Unique solution to f(x^2)^2 = (f(x)+1/f(x))/2, and f(0)=1, f'(0) nonzero.

G.f.: theta_3/theta_4 = (Sum_{k} x^k^2)/(Sum_{k} (-x)^k^2) = (Product_{k>0} (1-x^(4k-2))/((1-x^(4k-1))(1-x^(4k-3)))^2)^2.

G.f. A(x) satisfies 0=f(A(x), A(x^3)) where f(u, v)=(1-u^4)(1-v^4)-(1-uv)^4 . - Michael Somos Jan 01 2006

Expansion of phi(q) / phi(-q) = chi(q)^2 / chi(-q)^2 = psi(q)^2 / psi(-q)^2 = phi(-q^2)^2 / phi(-q)^2 = phi(q)^2 / phi(-q^2)^2 = chi(-q^2)^2 / chi(-q)^4 = chi(q)^4 / chi(-q^2)^2 = f(q)^2 / f(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A028939.

Expansion of Jacobian elliptic function 1/sqrt(k') in powers of q. - see Fine.

1 + 4*q + 8*q^2 + 16*q^3 + 32*q^4 + 56*q^5 + 96*q^6 + 160*q^7 + 256*q^8 + ...

(PARI) a(n)=local(A, B); if(n<0, 0, A=1+4*x; for(k=2, n, B=A+x^2*O(x^k); A+=Pol(2*subst(B, x, x^2)^2-B-1/B)/x/8); polcoeff(A, n)) /* Michael Somos Jul 07 2005*/

(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))^2, n))} /* Michael Somos Jan 01 2006 */

Cf. A014969, A001936, A001938, A079006, A127391, A127392.

A097243(n)=a(4n). 8*A022577(n)=a(4n+2). a(n)=4*A123655(n) if n>0.

Sequence in context: A048168 A131649 A003199 this_sequence A036313 A121986 A108569

Adjacent sequences: A007093 A007094 A007095 this_sequence A007097 A007098 A007099

nonn,easy

njas, Simon Plouffe (plouffe(AT)math.uqam.ca)