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A133988
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Expansion of phi(x) / chi(x^3) in powers of x where phi(), chi() are Ramanujan theta functions.
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2
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1, 2, 0, -1, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^2, b = -x. - Michael Somos, Jan 21 2012
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Quintuple Product Identity
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FORMULA
| Expansion of q^(-1/8) * eta(q^2)^5 * eta(q^3) * eta(q^12) / ( eta(q) * eta(q^4) * eta(q^6) )^2 in powers of q.
Expansion of psi(-x) + 3 * x * psi(-x^9) in powers of x where psi() is a Ramanujan theta function.
Expansion of f(x, x^5) * f(x) / f(-x^6) = f(-x^3, x^6) + x * f(1, -x^9) in powers of x where f() is a Ramanujan theta function.
Expansion of phi(x^9) / chi(x^3) + 2 * x * psi(-x^9) in powers of x where phi(), chi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 2, -3, 1, -1, 2, -2, 2, -1, 1, -3, 2, -1, ...].
a(n) = b(8*n + 1) where b(n) is multiplicative and b(3^(2e)) = -2 * (-1)^e, b(p^(2e)) = (-1)^e if p == 3, 5 (mod 8), b(p^(2e)) = +1 if p == 1, 7 (mod 8) and b(p^(2e-1)) = b(2^e) = 0 if e>0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 12 (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A133985.
G.f.: Sum_{k>0} (-1)^[k/2] * (x^((k^2 - k)/2) + 3 * x^(9*(k^2 - k)/2 + 1) ).
G.f.: Sum_{k} (-1)^(k + [k/2]) * x^(3*k * (3*k + 1) / 2) * ( x^(-3*k) + x^(3*k + 1) ).
A089812(n) = (-1)^n * a(n).
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EXAMPLE
| 1 + 2*x - x^3 + x^6 - 2*x^10 - x^15 - x^21 - 2*x^28 + x^36 - x^45 + ...
q + 2*q^9 - q^25 + q^49 - 2*q^81 - q^121 - q^169 - 2*q^225 + q^289 - ...
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PROG
| (PARI) {a(n) = (-1)^(n\3) * ((n + 1)%3) * issquare( 8*n + 1)}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^12 + A) / ( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) )^2, n))}
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CROSSREFS
| Cf. A089812, A133985.
Sequence in context: A165408 A171368 A089812 * A123858 A193261 A035145
Adjacent sequences: A133985 A133986 A133987 * A133989 A133990 A133991
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Oct 01 2007
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