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A143433
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Expansion of f(-x, x^3) in powers of x where f(,) is Ramanujan's two variable theta function.
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2
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1, -1, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 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, 1, 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, 1, 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
| 0,1
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (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
| Euler transform of period 16 sequence [ -1, 0, 1, 1, 1, -1, -1, -2, -1, -1, 1, 1, 1, 0, -1, -1, ...].
Pattern of signs of nonzero terms is A143431.
G.f.: Sum_{k>=0} (-1)^(k + [k/4]) * x^(k * (k+1) / 2).
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EXAMPLE
| 1 - x + x^3 - x^6 - x^10 + x^15 - x^21 + x^28 + x^36 - x^45 + x^55 - x^66 + ...
q - q^9 + q^25 - q^49 - q^81 + q^121 - q^169 + q^225 + q^289 - q^361 + ...
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PROG
| (PARI) {a(n) = if( n<0, 0, if(issquare(8*n + 1, &n), n = n\2; (-1)^(n + n\4), 0))}
(PARI) {a(n) = local(A); if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^( [1, 1, 0, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, 0, 1, 1] [k%16 + 1]), 1 + x * O(x^n)), n))}
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CROSSREFS
| (-1)^n * A143434(n) = a(n).
Sequence in context: A106459 * A143434 A197870 A033806 A033802 A033800
Adjacent sequences: A143430 A143431 A143432 * A143434 A143435 A143436
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Aug 14 2008
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