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

Author

Michael Somos, Aug 23 2005

Keywords

• nonn
• mult

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

Crossrefs

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

Programs

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 *)

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)))))}

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