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A143323
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Expansion of eta(q^2)^4 * eta(q^5) * eta(q^20)^2 / ( eta(q) * eta(q^4)^2 * eta(q^10)^2 ) in powers of q.
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2
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1, 1, -2, -1, 1, -2, -2, 1, 3, 1, 0, 2, 0, -2, -2, -1, 0, 3, 0, -1, 4, 0, -2, -2, 1, 0, -4, 2, 2, -2, 0, 1, 0, 0, -2, -3, 0, 0, 0, 1, 2, 4, -2, 0, 3, -2, -2, 2, 3, 1, 0, 0, 0, -4, 0, -2, 0, 2, 0, 2, 2, 0, -6, -1, 0, 0, -2, 0, 4, -2, 0, 3, 0, 0, -2, 0, 0, 0, 0, -1, 5, 2, -2, -4, 0, -2, -4, 0, 2, 3, 0, 2, 0, -2, 0, -2, 0, 3, 0, -1, 2, 0, -2, 0, 4
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Expansion of q * phi(-q^2) * chi(q) * psi(q^10) * chi(-q^5) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q * phi(-q^2)^2 * psi(-q^5)^2 / (f(-q) * f(-q^5)) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Apr 07 2015
Euler transform of period 20 sequence [ 1, -3, 1, -1, 0, -3, 1, -1, 1, -2, 1, -1, 1, -3, 0, -1, 1, -3, 1, -2, ...].
Multiplicative with a(2^e) = -(-1)^e unless e=0, a(p^e) = 1 if p=5, a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), a(p^e) = e+1 if p == 1, 9 (mod 20), a(p^e) = (e+1)(-1)^e if p == 3, 7 (mod 20).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 20^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A129391.
G.f.: Sum_{k>0} -(-1)^k F(x^(2*k - 1)) where F(x) = x * (1 + x) * (1 - x^2) / (1 + x^5).
G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2 * (1 - x^(5*k)) * ( 1 + x^(10*k))^2.
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EXAMPLE
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G.f. = q + q^2 - 2*q^3 - q^4 + q^5 - 2*q^6 - 2*q^7 + q^8 + 3*q^9 + q^10 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 2 q^(13/8) EllipticTheta[ 4, 0, q^2]^2 QPochhammer[ q^20]^2 / ( QPochhammer[ q] EllipticTheta[ 2, 0, q^(5/2)]), {q, 0, n}]; (* Michael Somos, Apr 07 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, (-1)^n * (qfrep([2, 1; 1, 3], n)[n] - qfrep([1, 0; 0, 5], n)[n] ))};
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^5 + A) * eta(x^20 + A)^2 / ( eta(x + A) * eta(x^4 + A)^2 * eta(x^10 + A)^2 ), n))};
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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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