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A094247
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Expansion of (phi(-q^5)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
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3
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1, -1, 0, -1, 1, 0, 0, -1, 1, -1, 0, 0, 2, 0, 0, -1, 2, -1, 0, -1, 0, 0, 0, 0, 1, -2, 0, 0, 2, 0, 0, -1, 0, -2, 0, -1, 2, 0, 0, -1, 2, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, -2, 2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, 2, 0, 0, -2, 0, 0, 0, -1, 2, -2, 0, 0, 0, 0, 0, -1, 1, -2, 0, 0, 2, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 2, -1, 0, -1, 2, 0, 0, -2
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OFFSET
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1,13
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COMMENTS
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LINKS
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FORMULA
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Expansion of q * f(q^5) * f(-q^20) * chi(-q) in powers of q where f() and chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^10)^3 / (eta(q^2) * eta(q^5)) in powers of q.
Euler transform of period 10 sequence [-1, 0, -1, 0, 0, 0, -1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = -1 if e > 0. a(5^e) = 1, a(p^e) = e+1 if p == 1, 5 (mod 8), a(p^e) = (1 + (-1)^e) / 2 if p == 3, 7 (mod 8).
G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 4 (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A214316. - Michael Somos, Jul 12 2012
G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(10*k))^3 / ((1 - x^(2*k)) * (1 - x^(5*k))).
G.f.: Sum_{k>0} Kronecker( -100, k) * x^k / (1 + x^k) = Sum_{k>0} Kronecker( -25, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)). - Michael Somos, Jul 12 2012
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EXAMPLE
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G.f. = q - q^2 - q^4 + q^5 - q^8 + q^9 - q^10 + 2*q^13 - q^16 + 2*q^17 - q^18 - q^20 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^5]^2 - EllipticTheta[ 4, 0, q]^2)/4, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^5] QPochhammer[ q^20] QPochhammer[q, q^2], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^3 / (eta(x^2 + A) * eta(x^5 + A)), n))};
(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, kronecker( -100, d)))}; /* Michael Somos, Aug 24 2006 */
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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