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A080332
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G.f.: Prod_{n>0} (1 - x^n)^3 * (1 - x^(2*n - 1))^2 = Sum_n (6*n + 1) * x^(n*(3*n + 1)/2).
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4
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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, 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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REFERENCES
| J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 306.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.6); p. 84, Eq. (32.63).
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266 MR0099904 (20 #6340)
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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 q^(-1/24) * eta(q)^5 / eta(q^2)^2 in powers of q.
G.f.: theta_4(x)^2 * (Sum_n (-1)^n * x^(n*(3*n + 1)/2)).
a(n)= b(24*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2* p^(e/2) if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 * (-p)^(e/2) if p == 5 (mod 6). - Michael Somos, May 26 2005
Euler transform of period 2 sequence [ -5, -3, ...]. - Michael Somos, Sep 09 2007
Expansion of f(-x)^2 * phi(x) = f(-x^2) * phi(-x^2)^2 in powers of x^2 where phi(), f() are Ramanujan theta functions. - Michael Somos, Feb 18 2003
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A113277. - Michael Somos, Feb 18 2003
a(5*n + 3) = a(5*n + 4) = a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0. a(25*n + 1) = -5 * a(n). - Michael Somos, Feb 18 2003
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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 + ...
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) = local(A, p, e); if( n<1, n==0, A = factor(24*n + 1); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( (p<5)|(e%2), 0, if( p%6==1, p, -p)^(e\2)))))} /* Michael Somos, May 26 2005 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^2 + A)^2, n))}
(PARI) {a(n) = if( issquare( 24*n + 1, &n), n * kronecker( -3, n), 0)}
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CROSSREFS
| Cf. A010815, A113277.
Sequence in context: A133079 * A134756 A178902 A176713 A011350 A161018
Adjacent sequences: A080329 A080330 A080331 * A080333 A080334 A080335
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KEYWORD
| sign,easy
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AUTHOR
| Michael Somos, Feb 18, 2003
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EXTENSIONS
| Definition changed by N. J. A. Sloane (njas(AT)research.att.com), Aug 14 2007
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