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A111973
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Expansion of ((eta(q^2)eta(q^4))^6/(eta(q)eta(q^8))^4-1)/4 in powers of q.
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2
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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
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OFFSET
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1,2
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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)))))}
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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_Michael Somos_, Aug 23 2005
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STATUS
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approved
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