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Search: id:A136748
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| A136748 |
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Expansion of (a(q) - a(q^2) - 4*a(q^4) + 4*a(q^8)) / 6 in powers of q where a() is a cubic AGM function. |
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2
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| 1, -1, 1, -3, 0, -1, 2, 3, 1, 0, 0, -3, 2, -2, 0, -3, 0, -1, 2, 0, 2, 0, 0, 3, 1, -2, 1, -6, 0, 0, 2, 3, 0, 0, 0, -3, 2, -2, 2, 0, 0, -2, 2, 0, 0, 0, 0, -3, 3, -1, 0, -6, 0, -1, 0, 6, 2, 0, 0, 0, 2, -2, 2, -3, 0, 0, 2, 0, 0, 0, 0, 3, 2, -2, 1, -6, 0, -2, 2, 0, 1, 0, 0, -6, 0, -2, 0, 0, 0, 0, 4, 0, 2, 0, 0, 3, 2, -3, 0, -3, 0, 0, 2, 6, 0
(list; graph; listen)
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OFFSET
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1,4
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FORMULA
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Expansion of eta(q) * eta(q^3) * eta(q^4)^4 * eta(q^24)^2 / ( eta(q^2) * eta(q^8) * eta(q^12) )^2 in powers of q.
Euler transform of period 24 sequence [ -1, 1, -2, -3, -1, 0, -1, -1, -2, 1, -1, -2, -1, 1, -2, -1, -1, 0, -1, -3, -2, 1, -1, -2, ...].
a(n) is multiplicative with a(2) = -1, a(2^e) = -3 * (-1)^e if e>1, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A123484.
a(3*n) = a(n). a(6*n+5) = 0.
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^2 * (1 + x^k + x^(2*k)) * (1 - x^(4*k) + x^(8*k))^2.
Moebius transform is period 24 sequence [ 1, -2, 0, -2, -1, 0, 1, 6, 2, -1, 0, 1, -2, 0, -6, -1, 0, 1, 2, 0, 2, -1, 0, ...].
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EXAMPLE
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q - q^2 + q^3 - 3*q^4 - q^6 + 2*q^7 + 3*q^8 + q^9 - 3*q^12 + 2*q^13 + ...
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PROGRAM
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(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3+A) * eta(x^4 + A)^4 * eta(x^24 + A)^2 / ( eta(x^2 + A) * eta(x^8+A) * eta(x^12+A) )^2, n))}
(PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, if( e<2, -1, -3 * (-1)^e), if(p==3, 1, if(p%6>1, !(e%2), (e+1)))))))}
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, ((d%2) -4 * (d%8==4)) * kronecker(-3, n/d)))}
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CROSSREFS
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A033762(n) = a(2*n+1) = -a(4*n+2). A122861(n) = a(3*n+1). -3 * A093829(n) = a(4*n). A112604(n) = a(4*n+1). A112605(n) = a(4*n+3). A097195(n) = a(6*n+1).
Adjacent sequences: A136745 A136746 A136747 this_sequence A136749 A136750 A136751
Sequence in context: A124027 A097610 A129555 this_sequence A049765 A014573 A067166
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Jan 22 2008
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