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Search: id:A116604
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| A116604 |
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Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q. |
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4
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| 1, -3, 2, 0, 1, 0, 2, -6, 2, 0, 0, 0, 3, -3, 2, 0, 0, 0, 2, -6, 2, 0, 2, 0, 1, -6, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, -9, 0, 0, 1, 0, 4, -6, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, -6, 2, 0, 2, 0, 1, -6, 4, 0, 0, 0, 0, -6, 2, 0, 0, 0, 4, -3, 2, 0, 2, 0, 2, -6, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, -6, 4, 0, 0, 0, 2, -12, 2, 0, 0, 0, 4, 0, 0
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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G.f.: Product_{k>0} (1-x^k)^2*(1+x^(2k))*(1-x^k+x^(2k))*(1+x^(6k)).
G.f.: Sum_{k>=0} x^(3k)/(1+x^(6k+1)) -2*x^(3k+1)/(1+x^(6k+3)) + x^(3k+2)/(1+x^(6k+5)).
Expansion of psi(q^2)^2 - 3 * q * psi(q^6)^2 in powers of q where psi() is a Ramanujan theta function.
Euler transform of period 12 sequence [ -3, -1, -2, -2, -3, 0, -3, -2, -2, -1, -3, -2, ...].
Moebius transform is period 24 sequence [ 1, -1, -4, 0, 1, 4, -1, 0, 4, -1, -1, 0, 1, 1, -4, 0, 1, -4, -1, 0, 4, 1, -1, 0, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = -1 + 2 * (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 12 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A121450.
a(6*n + 3) = a(6*n + 5) = 0.
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EXAMPLE
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q - 3*q^3 + 2*q^5 + q^9 + 2*q^13 - 6*q^15 + 2*q^17 + 3*q^25 - 3*q^27 + ...
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PROGRAM
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(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker(-4, n/d) * [ -2, 1, 1][d%3 + 1]))}
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 2*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==3, -1 + 2 * (-1)^e, if(p%12 < 6, e+1, (1 + (-1)^e) / 2)))))) }
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)), n))}
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CROSSREFS
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A002175(n)=a(6n). A008441(n)=a(2n).
Sequence in context: A059033 A133209 A131290 this_sequence A138741 A079618 A117406
Adjacent sequences: A116601 A116602 A116603 this_sequence A116605 A116606 A116607
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KEYWORD
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sign
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AUTHOR
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Michael Somos Feb 18 2006, Apr 03 2008
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