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Search: id:A113277
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| A113277 |
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Expansion of q^(-1/3) * eta(q^2)^5 / eta(q)^2 in powers of q. |
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3
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| 1, 2, 0, 0, 0, -4, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -17, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)
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FORMULA
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a(n) = b(3n+1) where b(n) is multiplicative and a(p^e) = 0 if e is odd, a(3^e) = 0^e, a(2^e) = -(-2)^(e/2), a(p^e) = p^(e/2) if p == 1 (mod 3), a(p^e) = (-p)^(e/2) if p == 2 (mod 3).
Euler transform of period 2 sequence [2, -3, ...].
G.f.: Sum_{k} (3k+1)(-x)^(3k^2+2k) = Product_{k>0} (1-x^k)^3(1+x^k)^5.
Expansion of psi(q^2) * f(q)^2 = phi(q) * f(-q^4)^2 in powers of q where phi(), psi(), f() are Ramanujan theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3456^(1/2) (t/i)^(3/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A080332.
a(4*n+2) = a(4*n+3) = a(5*n+2) = a(5*n+4) = a(8*n+4) = 0. a(25*n+8) = -5 * a(n).
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EXAMPLE
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q + 2*q^4 - 4*q^16 - 5*q^25 + 7*q^49 + 8*q^64 - 10*q^100 - 11*q^121 +...
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PROGRAM
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(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^5/eta(x+A)^2, n))}
(PARI) a(n)=if(issquare(3*n+1, &n), n*(-1)^(n%3+n), 0)
(PARI) {a(n)=local(A, p, e); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(e%2, 0, (-1)^(p==2)*(-(-1)^(p%3)*p)^(e/2)))))}
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CROSSREFS
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Cf. A114855(n) = (-1)^n * a(n). 2 * A114855(n) = a(4*n+1). A080332(n) = a(8*n).
Sequence in context: A134309 A051516 A127391 this_sequence A114855 A100951 A011991
Adjacent sequences: A113274 A113275 A113276 this_sequence A113278 A113279 A113280
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Oct 21 2005
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