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Search: id:A030202
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| A030202 |
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Expansion of eta(q)*eta(q^5)/q^(1/4) in powers of q. |
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+0
2
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| 1, -1, -1, 0, 0, 0, 1, 2, 0, 0, -2, 1, -1, 0, 0, -2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, -2, 0, 0, 1, -1, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 2, 0, 0, -2, 1, 0, 0, 0, -2, 0, -2, 0, 0, -2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, -2, 0, 0, 2, -1, -2, 0, 0
(list; graph; listen)
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OFFSET
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0,8
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COMMENT
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Euler transform of period 5 sequence [ -1,-1,-1,-1,-2,...].
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REFERENCES
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Bruce Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; page 44.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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FORMULA
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Expansion of f(-q, -q^4) * f(-q^2, -q^3) in powers of q where f() is the Ramanujan two variable theta function.
Expansion of q^(-1) * (phi(q) * phi(q^20) - phi(q^4) * phi(q^5)) / 2 in powers of q^4 where phi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 80^(1/2) (t/i) f(t) where q = exp(2 pi i t).
a(n) = b(4n+1) where b(n) is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, b(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), b(p^e) = (i^n +(-i)^n)/2 if p == 3, 7 (mod 20), b(p^e) = (-1)^(e*y) * (e+1) if p == 1, 9 (mod 20) where p = x^2 + 5*y^2. - Michael Somos Sep 04 2007
G.f.: Product_{k>0} (1-x^k) * (1-x^(5*k)).
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EXAMPLE
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q - q^5 - q^9 + q^25 + 2*q^29 - 2*q^41 + q^45 - q^49 - 2*q^61 + q^81 + ...
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PROGRAM
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(PARI) {a(n) = if( n<0, 0, polcoeff( eta(x^5 + x*O(x^n)) * eta(x + x*O(x^n)), n))} /* Michael Somos Sep 04 2007 */
(PARI) {a(n)= local(A, p, e, x, y); if(n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if(p==2, 0, if(p==5, (-1)^e, if(p%20>10, !(e%2), if(p%4==3, kronecker(-4, e+1), for(y=1, sqrtint(p\5), if(issquare(p-5*y^2), x=y; break)); (-1)^(e*x) *(e+1))))))))} /* Michael Somos Sep 04 2007 */
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CROSSREFS
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Sequence in context: A035437 A156996 A029304 this_sequence A159818 A081827 A100286
Adjacent sequences: A030199 A030200 A030201 this_sequence A030203 A030204 A030205
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KEYWORD
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sign,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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