login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A111973 Expansion of ((eta(q^2)eta(q^4))^6/(eta(q)eta(q^8))^4-1)/4 in powers of q. 2

%I #13 Aug 22 2023 11:58:36

%S 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,

%T 40,48,30,48,32,6,48,36,48,78,38,40,56,36,42,64,44,72,78,48,48,24,57,

%U 62,72,84,54,80,72,48,80,60,60,144,62,64,104,6,84,96,68,108,96,96,72

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

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

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

%H Amiram Eldar, <a href="/A111973/b111973.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F 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

%t 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 *)

%o (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]))

%o (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))}

%o (PARI) a(n)= local(x); if(n<1, 0, x=2^valuation(n,2); sigma(n/x)*if(x>2,6,x))

%o (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)))))}

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

%K nonn,mult

%O 1,2

%A _Michael Somos_, Aug 23 2005

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)