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Search: id:A134177
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| A134177 |
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Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^3) * eta(q^8) * eta(q^12)^4 / ( eta(q) * eta(q^4)^3 * eta(q^6)^3 * eta(q^24) ) in powers of q. |
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1
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| 1, 1, -2, -2, 1, 2, 0, -2, 0, 0, -2, 0, 3, 1, -2, -2, 2, 4, 0, 0, 0, 0, -2, 0, 3, 0, -2, -4, 0, 2, 0, -2, 0, 0, 0, 0, 2, 3, -4, -2, 1, 2, 0, -2, 0, 0, -2, 0, 2, 2, -2, -2, 4, 2, 0, 0, 0, 0, 0, 0, 3, 0, -4, -2, 0, 2, 0, -2, 0, 0, 0, 0, 4, 3, -2, -2, 0, 4, 0, -2, 0, 0, -4, 0, 1, 0, -2, -6, 2, 2, 0, 0, 0, 0, -2, 0, 2, 0, -2, -2, 0, 4, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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Expansion of phi(-q^3) * psi(q^4) + q * phi(-q) * psi(q^12) in powers of q where phi(), psi() are Ramanujan theta functions.
a(n) = b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 11 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 7 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ 1, -3, 0, 0, 1, -1, 1, -1, 0, -3, 1, -2, 1, -3, 0, -1, 1, -1, 1, 0, 0, -3, 1, -2, ...].
G.f. is a period 1 Fourier series that satisfies f(-1 / (96 t)) = 96^(1/2) (t/i) f(t) where q = exp(2 pi i t).
a(12n+6) = a(12n+8) = a(12n+9) = a(12n+11) = 0.
G.f.: Sum_{k>=0} a(k) * x^(2*k+1) = Sum_{k>0} kronecker( -2, k) * (x^k - x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 + x^(6*k)) * (1 - x^(2*k) + x^(4*k))^2 / (1 - x^(4*k) + x^(8*k)).
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EXAMPLE
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q + q^3 - 2*q^5 - 2*q^7 + q^9 + 2*q^11 - 2*q^15 - 2*q^21 + 3*q^25 + 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( 12, d)* kronecker( -2, n/d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A)^4 / ( eta(x + A) * eta(x^4 + A)^3 * eta(x^6 + A)^3* eta(x^24 + A) ), n))}
(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, if(p%24>12, !(e%2), (e+1)*kronecker(3, p)^e))))))}
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CROSSREFS
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(-1)^n * A128580(n) = a(n). A113780(n)= a(12n).
Sequence in context: A094022 A128580 A129402 this_sequence A104405 A089077 A130071
Adjacent sequences: A134174 A134175 A134176 this_sequence A134178 A134179 A134180
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Oct 11 2007
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