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A140727
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Expansion of (phi(q) * phi(q^15) - phi(q^3) * phi(q^5)) / 2 in powers of q where phi() is a Ramanujan theta function..
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1
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1, 0, -1, 1, -1, 0, 0, -2, 1, 0, 0, -1, 0, 0, 1, 3, -2, 0, 2, -1, 0, 0, -2, 2, 1, 0, -1, 0, 0, 0, 2, -4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, -2, -3, 1, 0, 2, 0, -2, 0, 0, 0, -2, 0, 0, 1, 2, 0, 0, 5, 0, 0, 0, -2, 2, 0, 0, -2, 0, 0, -1, 2, 0, 0, 2, -3, 1, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, -2, -2, 0, -2, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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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
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(v).
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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 q * f(-q^2) * f(-q^30) * chi(q^3) * chi(q^5) in powers of q where f(), chi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^6)^2 * eta(q^10)^2 * eta(q^30) / (eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20)) in powers of q.
Euler transform of period 60 sequence [ 0, -1, 1, -1, 1, -2, 0, -1, 1, -2, 0, -1, 0, -1, 2, -1, 0, -2, 0, -1, 1, -1, 0, -1, 1, -1, 1, -1, 0, -4, 0, -1, 1, -1, 1, -1, 0, -1, 1, -1, 0, -2, 0, -1, 2, -1, 0, -1, 0, -2, 1, -1, 0, -2, 1, -1, 1, -1, 0, -2, ...].
a(n) is multiplicative with a(2^e) = (-1)^e * (e-1) if e>0. a(3^e) = a(5^e) = (-1)^e, a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (-1)^e * (e+1) if p == 2, 8 (mod 15), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15).
G.f. is a period 1 Fourier series which satisfies f( -1 / (60 t)) = 60^(1/2) (t/i) f(t) where q = exp(2 pi i t).
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EXAMPLE
| q - q^3 + q^4 - q^5 - 2*q^8 + q^9 - q^12 + q^15 + 3*q^16 - 2*q^17 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (-1)^(n + d) * kronecker(5, d) * kronecker(-3, n/d)))}
(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, (-1)^e * (e-1), if( p==3 | p==5, (-1)^e, if( kronecker(p, 15)==1, (e+1) * (-1)^(e*valuation(p%15, 2)), (1 + (-1)^e) / 2))))))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^2 * eta(x^10 + A)^2 * eta(x^30 + A) / (eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A)), n))}
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CROSSREFS
| -(-1)^n * A140728(n) = a(n). A122855(n) = |a(n)|.
Sequence in context: A060398 A122855 * A140728 A130068 A051699 A007920
Adjacent sequences: A140724 A140725 A140726 * A140728 A140729 A140730
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, May 29 2008
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