A111973 Expansion of ((eta(q^2)eta(q^4))^6/(eta(q)eta(q^8))^4-1)/4 in powers of q.

1, 2, 4, 6, 6, 8, 8, 6, 13, 12, 12, 24, 14, 16, 24, 6, 18, 26, 20, 36, 32, 24, 24, 24, 31, 28, 40, 48, 30, 48, 32, 6, 48, 36, 48, 78, 38, 40, 56, 36, 42, 64, 44, 72, 78, 48, 48, 24, 57, 62, 72, 84, 54, 80, 72, 48, 80, 60, 60, 144, 62, 64, 104, 6, 84, 96, 68, 108, 96, 96, 72

Offset

1,2

References

Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373, Entry 31.

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.29).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(2)=2, a(2^e)=6 if e>1, a(p^e)=(p^(e+1)-1)/(p-1) if p>2.

G.f.: ((theta_3(q)theta_3(q^2))^2-1)/4 where theta_3(q)=1+2(q+q^4+q^9+...).

G.f.: Sum_{k>0} 2*x^(4k)/(1+x^(4k))^2 +x^(2k-1)/(1-x^(2k-1))^2 = Sum_{k>0} +(2+(-1)^k)k x^(2k)/(1+x^(2k)) +(2k-1)x^(2k-1)/(1-x^(2k-1)). - Michael Somos, Oct 22 2005

Mathematica

f[p_, e_] := (p^(e+1)-1)/(p-1); f[2, 1] = 2; f[2, e_] := 6; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 22 2023 *)

Prog

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, d*(-1)^((d+1)*(n/d+1))*[2, 1, 0, 1][n/d%4+1]))
(PARI) {a(n)= local(A); if(n<1, 0, A=x*O(x^n); polcoeff( ((eta(x^2+A)*eta(x^4+A))^6/(eta(x+A)*eta(x^8+A))^4-1)/4, n))}
(PARI) a(n)= local(x); if(n<1, 0, x=2^valuation(n, 2); sigma(n/x)*if(x>2, 6, x))
(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 2+4*(e>1), (p^(e+1)-1)/(p-1)))))}

Crossrefs

Cf. A097057(n)=4*a(n), if n>0.

Sequence in context: A131450 A114218 A133691 * A349787 A161655 A092517

Adjacent sequences: A111970 A111971 A111972 * A111974 A111975 A111976

Keyword

nonn,mult

Author

Michael Somos, Aug 23 2005

Status

approved