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A134014
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Expansion of phi(-q) * phi(q^4) in powers of q where phi() is a Ramanujan theta function.
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2
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1, -2, 0, 0, 4, -4, 0, 0, 4, -2, 0, 0, 0, -4, 0, 0, 4, -4, 0, 0, 8, 0, 0, 0, 0, -6, 0, 0, 0, -4, 0, 0, 4, 0, 0, 0, 4, -4, 0, 0, 8, -4, 0, 0, 0, -4, 0, 0, 0, -2, 0, 0, 8, -4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 4, -8, 0, 0, 8, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 8, -2, 0, 0, 0, -8, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 12, -4, 0, 0, 8
(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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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of eta(q)^2 * eta(q^8)^5 / ( eta(q^2) * eta(q^4)^2 * eta(q^16)^2 ) in powers of q.
Euler transform of period 16 sequence [ -2, -1, -2, 1, -2, -1, -2, -4, -2, -1, -2, 1, -2, -1, -2, -2, ...].
Moebius transform is period 16 sequence [ -2, 2, 2, 4, -2, -2, 2, 0, -2, 2, 2, -4, -2, -2, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A134013.
a(4*n+2) = a(4*n+3) = 0.
G.f.: 1 - 2 * ( x/(1+x^2) + x^3/(1+x^6) - 2 * x^4/(1+x^8) + ... ).
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EXAMPLE
| 1 - 2*q + 4*q^4 - 4*q^5 + 4*q^8 - 2*q^9 - 4*q^13 + 4*q^16 - 4*q^17 + ...
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PROG
| (PARI) {a(n) = if( n<1, n==0, if( n%4 < 2, (n%2*-6 + 4) * sumdiv(n, d, kronecker(-4, d))))}
(PARI) {a(n) = (-1)^n * if( n<1, n==0, 2 * qfrep([1, 0; 0, 4], n)[n])}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A)^5 / eta(x^2 + A) / eta(x^4 + A)^2 / eta(x^16 + A)^2, n))}
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CROSSREFS
| (-1)^n * A004531(n) = a(n). -2 * A134015(n) = a(n) unless n=0. A004018(n) = a(4*n). - A004020(n) = a(4*n+1).
Sequence in context: A072740 A080964 A004531 * A072071 A045836 A072070
Adjacent sequences: A134011 A134012 A134013 * A134015 A134016 A134017
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Oct 02 2007
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