|
| |
|
|
A128771
|
|
Expansion of phi(-q)/phi(-q^9) in powers of q where phi() is a Ramanujan theta function.
|
|
2
| |
|
|
1, -2, 0, 0, 2, 0, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, 2, 0, 0, -8, 0, 0, 8, 0, 0, 2, 0, 0, -16, 0, 0, 16, 0, 0, 4, 0, 0, -28, 0, 0, 28, 0, 0, 8, 0, 0, -48, 0, 0, 46, 0, 0, 12, 0, 0, -80, 0, 0, 76, 0, 0, 20, 0, 0, -126, 0, 0, 120, 0, 0, 32, 0, 0, -196, 0, 0, 184, 0, 0, 48, 0, 0, -300, 0, 0, 280, 0, 0, 72, 0, 0
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
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).
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
FORMULA
| Expansion of eta(q)^2* eta(q^18)/( eta(q^2)* eta(q^9)^2 ) in powers of q.
Euler transform of period 18 sequence [ -2, -1, -2, -1, -2, -1, -2, -1, 0, -1, -2, -1, -2, -1, -2, -1, -2, 0, ...].
G.f. A(x) satisfies 0= f(A(x), A(x^2)) where f(u, v)= (1-u)* (u-v^2) -2*u* (v-1).
G.f. A(x) satisfies 0= f(A(x), A(x^3)) where f(u, v)= (u-v)^3 -u* (3-u)* (v-1)* (3 -2*u +u*v).
G.f.: Product_{k>0} (1-x^k)* (1+x^(9k))/( (1+x^k)* (1-x^(9k)) ).
a(3n+2)= a(3n+3)= 0.
Empirical : sum(exp(-Pi/3)^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = sqrt(3). Simon Plouffe, Feb. 20, 2011.
|
|
|
PROG
| (PARI) {a(n)= local(A); if(n<0, 0, A=x*O(x^oo); polcoeff( eta(x+A)^2* eta(x^18+A)/ eta(x^2+A)/ eta(x^9+A)^2, n))}
|
|
|
CROSSREFS
| Convolution inverse of A128770. -2*A092848(n)= a(3n+1).
Sequence in context: A193531 A093492 * A139380 A000122 A002448 A033759
Adjacent sequences: A128768 A128769 A128770 * A128772 A128773 A128774
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Michael Somos, Mar 27 2007
|
| |
|
|
