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A133985
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Expansion of f(-x, x^2) in powers of x where f() is Ramanujan's two variable theta function.
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2
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1, -1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is a number of A001318.
The exponents in the q-series for this sequence are the squares of the numbers of A007310.
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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 phi(x^3) / chi(x) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/24) * eta(q) * eta(q^4) * eta(q^6)^5 / ( eta(q^2) * eta(q^3) * eta(q^12) )^2 in powers of q.
Euler transform of period 12 sequence [ -1, 1, 1, 0, -1, -2, -1, 0, 1, 1, -1, -1, ...].
a(n) = b(24*n + 1) where b(n) is multiplicative with 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) = b(3^e) = 0 if e>0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 4 (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A133988.
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = -a(n). a(n) = (-1)^n * A080995(n).
G.f. Sum_{k>=0} a(k) * q^(24*k + 1) = Sum_{k} (-1)^[k/2] q^(6*k + 1)^2.
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EXAMPLE
| 1 - x + x^2 - x^5 - x^7 + x^12 - x^15 + x^22 + x^26 - x^35 + x^40 + ...
q - q^25 + q^49 - q^121 - q^169 + q^289 - q^361 + q^529 + q^625 + ...
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PROG
| (PARI) {a(n) = (-1)^n * issquare( 24*n + 1) }
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^5 / ( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) )^2, n))}
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CROSSREFS
| Cf. A001318, A007310, A080995, A133988.
Sequence in context: A121373 * A143062 A199918 A074910 A115356 A115359
Adjacent sequences: A133982 A133983 A133984 * A133986 A133987 A133988
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Oct 01 2007, Oct 04 2007
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