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Search: id:A114855
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| A114855 |
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Expansion of q^(-1/3) * (eta(q) * eta(q^4))^2 / eta(q^2) in powers of q. |
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2
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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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S. Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 147, Ed. G. H. Hardy et al., AMS Chelsea 2000.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266 MR0099904 (20 #6340)
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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(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 4 sequence [ -2, -1, -2, -3, ...].
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=(u*w*(u+2*w)*(u+4*w))^2 -v^6*(u^2+4*u*w+8*w^2).
G.f.: Sum_{k} (3k+1) x^(3k^2+2k) = Product_{k>0} (1-x^k)^2*(1+x^(2k))*(1-x^(4k)).
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.
a(4*n+2) = a(4*n+3) = a(8*n+4) = 0. a(4*n+1) = -2*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)=if(issquare(3*n+1, &n), n*-(-1)^(n%3), 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%3)*p)^(e/2)))))}
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^4+A))^2/ eta(x^2+A), n))}
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CROSSREFS
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A113277(n) = (-1)^n * a(n). A080332(n) = a(8*n).
Sequence in context: A051516 A127391 A113277 this_sequence A100951 A011991 A129183
Adjacent sequences: A114852 A114853 A114854 this_sequence A114856 A114857 A114858
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Jan 01 2006
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