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Search: id:A111932
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| A111932 |
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Expansion of (1/3)* (b(q^2)^2/b(q))* (c(q^2)^2/c(q)) in powers of q where b(), c() are cubic AGM analog functions. |
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2
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| 1, 2, 1, 4, 6, 2, 8, 8, 1, 12, 12, 4, 14, 16, 6, 16, 18, 2, 20, 24, 8, 24, 24, 8, 31, 28, 1, 32, 30, 12, 32, 32, 12, 36, 48, 4, 38, 40, 14, 48, 42, 16, 44, 48, 6, 48, 48, 16, 57, 62, 18, 56, 54, 2, 72, 64, 20, 60, 60, 24, 62, 64, 8, 64, 84, 24, 68, 72, 24, 96, 72, 8, 74, 76, 31
(list; graph; listen)
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OFFSET
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1,2
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223 Entry 3(iii).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 87, Eq. (33.2).
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FORMULA
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Multiplicative with a(2^e) = 2^e, a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p>3.
Euler transform of period 6 sequence [2, -2, 4, -2, 2, -4, ...].
Expansion of q(psi(q)psi(q^3))^2 in powers of q where psi(q) is a Ramanujan theta function.
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u*w*(u-4*v) -v*(v-4*w)^2.
G.f.: Sum_{k>0} k x^k(1-x^(2k))^2/(1-x^(6k)) = x Product_{k>0} ((1+x^k)(1+x^(3k)))^4((1-x^k)(1-x^(3k)))^2.
a(3n)=a(n), a(2n)=2a(n).
Expansion of (eta(q^2)* eta(q^6))^4/ (eta(q)* eta(q^3))^2 in powers of q.
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PROGRAM
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(PARI) a(n)=if(n<1, 0, sumdiv(n, d, (n/d%2)*d*(d%3>0)))
(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, p^e, if(p==3, 1, (p^(e+1)-1)/(p-1)))))) }
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^6+A))^4/ (eta(x+A)*eta(x^3+A))^2, n))}
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CROSSREFS
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Sequence in context: A059575 A120769 A165604 this_sequence A121456 A127535 A105364
Adjacent sequences: A111929 A111930 A111931 this_sequence A111933 A111934 A111935
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KEYWORD
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nonn,mult
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AUTHOR
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Michael Somos, Aug 21 2005, Apr 18 2007
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