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A138951
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Expansion of eta(q^2)^7 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^3 * eta(q^6)^3) in powers of q.
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1
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1, 2, -2, -6, -2, 4, 6, 0, -2, 2, -4, 0, 6, 4, 0, -12, -2, 4, -2, 0, -4, 0, 0, 0, 6, 6, -4, -6, 0, 4, 12, 0, -2, 0, -4, 0, -2, 4, 0, -12, -4, 4, 0, 0, 0, 4, 0, 0, 6, 2, -6, -12, -4, 4, 6, 0, 0, 0, -4, 0, 12, 4, 0, 0, -2, 8, 0, 0, -4, 0, 0, 0, -2, 4, -4, -18, 0, 0, 12, 0, -4, 2, -4, 0, 0, 8, 0, -12, 0, 4, -4, 0, 0, 0, 0, 0, 6, 4, -2, 0, -6
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| L.-C. Shen, On the Modular Equations of Degree 3. Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1108, Eq. (3.24).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of (3 * phi(-q^3)^2 - phi(-q)^2) / 2 in powers of q where phi() is a Ramanujan theta function.
Expansion of phi(q) * phi(-q^2) * chi(-q^3) / chi(q^3) in powers of q where phi(), chi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 2, -5, 0, -2, 2, -4, 2, -2, 0, -5, 2, -2, ...].
Moebius transform is period 24 sequence [ 2, -4, -8, 0, 2, 16, -2, 0, 8, -4, -2, 0, 2, 4, -8, 0, 2, -16, -2, 0, 8, 4, -2, 0, ...].
a(n) = 2 * b(n) where b(n) is multiplicative and b(2^e) = -1 if e>0, 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 / (12 t)) = (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A129447.
a(12*n + 7) = a(12*n + 11) = 0.
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 - x^(2*k) + x^(4*k)) / ((1 + x^(2*k))^2 * (1 - x^k + x^(2*k))^2).
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EXAMPLE
| 1 + 2*q - 2*q^2 - 6*q^3 - 2*q^4 + 4*q^5 + 6*q^6 - 2*q^8 + 2*q^9 - 4*q^10 + ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, -2 * (-1)^n * sumdiv(n, d, kronecker(-4, n/d) * [ -2, 1, 1][d%3 + 1]))}
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, -1, 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^2 + A)^7 * eta(x^3 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^6 + A)^3), n))}
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CROSSREFS
| (-1)^n * A138949(n) = a(n).
Sequence in context: A021446 A062401 A138949 * A163370 A071796 A121699
Adjacent sequences: A138948 A138949 A138950 * A138952 A138953 A138954
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Apr 03 2008
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