|
Search: id:A080332
|
|
|
| A080332 |
|
G.f.: Prod_{n>0} (1-x^n)^3(1-x^(2n-1))^2 = Sum_n (6n+1)x^(n(3n+1)/2). |
|
+0
4
|
|
| 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; listen)
|
|
|
OFFSET
|
0,2
|
|
|
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)
|
|
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(3n+1)/2)).
a(n)= b(24n+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(-q)^2 * phi(q) = f(-q^2) * phi(-q^2)^2 in powers of q^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
|
|
EXAMPLE
|
q - 5*q^25 + 7*q^49 - 11*q^121 + 13*q^169 - 17*q^289 + 19*q^361 +...
|
|
PROGRAM
|
(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)
|
|
CROSSREFS
|
Cf. A010815.
Sequence in context: A001111 A116916 A133079 this_sequence A134756 A011350 A161018
Adjacent sequences: A080329 A080330 A080331 this_sequence A080333 A080334 A080335
|
|
KEYWORD
|
sign,easy
|
|
AUTHOR
|
Michael Somos, Feb 18, 2003
|
|
EXTENSIONS
|
Definition changed by N. J. A. Sloane (njas(AT)research.att.com), Aug 14 2007
|
|
|
Search completed in 0.002 seconds
|
