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A132970
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Expansion of phi(-q) * chi(-q) in powers of q where phi(), chi() are Ramanujan theta functions.
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1
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1, -3, 2, -1, 5, -5, 3, -5, 6, -10, 10, -8, 13, -15, 15, -16, 23, -27, 25, -30, 35, -40, 42, -45, 55, -66, 68, -70, 86, -95, 100, -110, 125, -141, 150, -161, 185, -207, 215, -235, 266, -293, 310, -335, 375, -410, 438, -470, 521, -575, 610, -653, 725, -785, 835, -900, 983, -1070, 1140, -1220, 1331
(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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REFERENCES
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 60, Eqs. (26.64),(26.65),(26.66)
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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(-q) + 2 * psi(-q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.
Expansion of q^(1/24) * eta(q)^3 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -3, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 48^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A085140.
G.f.: ( Sum_{k} (-1)^k * x^k^2 ) / ( Product_{k>0} (1 + x^k) ).
G.f.: Product_{k>0} (1 - x^k) / (1 + x^k)^2.
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EXAMPLE
| 1/q - 3*q^23 + 2*q^47 - q^71 + 5*q^95 - 5*q^119 + 3*q^143 - 5*q^167 + ...
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PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + x*O(x^n)) * sum(k=1, sqrtint(n), 2 * x^k^2, 1), n))}
(PARI) {a(n) = local(A) ; if( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^3 / eta(x^2 + A)^2, n))}
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CROSSREFS
| a(n) = (-1)^n * A132969(n). A124226(n) = a(n) unless n=1.
Sequence in context: A091597 A091595 A132969 * A192022 A139377 A110712
Adjacent sequences: A132967 A132968 A132969 * A132971 A132972 A132973
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 04 2007
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