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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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).

Euler transform of period 4 sequence [4, -2, 4, 0, ...]. - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 22 2005

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

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 */

Self-convolution of A080054. - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 22 2005

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.

Convolution square of A080054.

Sequence in context: A048168 A131649 A003199 this_sequence A036313 A121986 A145108

Adjacent sequences: A007093 A007094 A007095 this_sequence A007097 A007098 A007099

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)