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A116916
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Expansion of q^(-1/8) * (eta(q)^3 + 3 * eta(q^9)^3) in powers of q^3.
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5
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1, 5, -7, 0, 0, -11, 0, 13, 0, 0, 0, 0, 17, 0, 0, -19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -35, 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, -43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 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).
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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 f(-x) * a(x) in powers of x where f() is a Ramnaujan theta function and a() is a cubic AGM theta function.
Expansion of f(-x)^3 + 3 * x * f(-x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 4608^(1/2) (t / i)^(3/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A202394.
G.f.: Sum_{k} (-1)^k * (6*k + 1) * x^(k * (3*k + 1) / 2).
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = 5 * a(n). a(n) = A010816(3*n).
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EXAMPLE
| 1 + 5*x - 7*x^2 - 11*x^5 + 13*x^7 + 17*x^12 - 19*x^15 - 23*x^22 + 25*x^26 + ...
q + 5*q^25 - 7*q^49 - 11*q^121 + 13*q^169 + 17*q^289 - 19*q^361 +...
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PROG
| (PARI) {a(n) = if( issquare( 24*n + 1, &n), n * kronecker( 3, n) * kronecker( -3, n))}
(PARI) {a(n) = if( n<1, n==0, n*=3; polcoeff( eta(x + x*O(x^n))^3 + 3 * x * eta(x^9 + O(x^n))^3, n))}
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CROSSREFS
| Cf. A010816, A202394.
Sequence in context: A048658 A001111 * A133079 A080332 A134756 A178902
Adjacent sequences: A116913 A116914 A116915 * A116917 A116918 A116919
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Feb 26 2006
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