1,5

Expansion of eta(q^8)^4/eta(q^4)^2 in powers of q.

Euler transform of period 8 sequence [0,0,0,2,0,0,0,-2,...]. - Michael Somos Apr 24 2004

a(n)=0 unless n=4k+1 in which case a(n) is the difference between number of divisors of n of form 4k+1 and 4k+3.

Multiplicative with a(2^e)=0 if e>0, a(p^e)=(1+(-1)^e)/2 if p==3 mod 4 otherwise a(p^e)=1+e. - Michael Somos Sep 18 2004

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; p. 78, Eq. (32.26).

T. D. Noe, Table of n, a(n) for n = 1..1000

Fine gives an explicit formula for a(n) in terms of the divisors of n.

a(n) = number of divisors of n of form 8n+1, 8n+5, 8n+6 minus number of divisors of form 8n+2, 8n+3, 8n+7. [I think Fine's version is simpler - njas]

G.f. = s(8)^4/(s(4)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]

Moebius transform is period 8 sequence [1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos Sep 02 2005

G.f.: Sum_{k>0} kronecker(-4, k) x^k/(1-x^(2k)) = Sum_{k>0} x^(2k-1)/(1+x^(4k-2)) . - Michael Somos Sep 20 2005

G.f.: Sum_{k>0} x^k(1-x^k)(1-x^(2k))(1-x^(3k))/(1-x^(8k)) = x Product_{k>0} (1-x^(8k))^4/(1-x^(4k))^2. - Michael Somos Apr 24 2004

Moebius transform is period 8 sequence [1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos, Sep 02 2005

(PARI) a(n)=if(n<1|n%4!=1, 0, sumdiv(n, d, (d%4==1)-(d%4==3))) /* Michael Somos Apr 24 2004 */

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, [0, 1, -1, -1, 0, 1, 1, -1][d%8+1])) /* Michael Somos Apr 24 2004 */

(PARI) a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff(eta(x^8+A)^4/eta(x^4+A)^2, n)) /* Michael Somos Apr 24 2004 */

A008441(n)=a(4n+1).

Sequence in context: A067618 A055029 A126812 this_sequence A086076 A085981 A127324

Adjacent sequences: A008439 A008440 A008441 this_sequence A008443 A008444 A008445

nonn,mult

njas