|
| |
|
|
A139215
|
|
Expansion of q^(-1) * psi(q) * phi(q^9) / (psi(q^3) * psi(q^6)) in power of q where phi(), psi() are Ramanujan theta functions.
|
|
2
| |
|
|
1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, -3, 0, -4, 0, 0, 0, 4, 0, 5, 0, 0, 0, -7, 0, -8, 0, 0, 0, 12, 0, 14, 0, 0, 0, -17, 0, -20, 0, 0, 0, 24, 0, 28, 0, 0, 0, -36, 0, -40, 0, 0, 0, 52, 0, 56, 0, 0, 0, -71, 0, -80, 0, 0, 0, 96, 0, 109, 0, 0, 0, -133, 0, -144, 0, 0, 0, 182, 0, 198, 0, 0, 0, -240, 0, -268
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| -1,11
|
|
|
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)^2 * eta(q^3) * eta(q^18)^5 / (eta(q) * eta(q^6) * eta(q^9)^2 * eta(q^12)^2 * eta(q^36)^2) in powers of q.
Euler transform of period 36 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, 2, -1, 1, 1, 1, -1, 0, -1, 1, -4, 1, -1, 0, -1, 1, 1, 1, -1, 2, -1, 1, -1, 1, -1, 0, -1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 pi i t) and g() is g.f. for A139213.
a(2*n) = 0 unless n=0.
|
|
|
EXAMPLE
| 1/q + 1 - q^3 + 2*q^9 + 2*q^11 - 3*q^15 - 4*q^17 + 4*q^21 + 5*q^23 + ...
|
|
|
PROG
| (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)^5 / (eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2 * eta(x^12 + A)^2 * eta(x^36 + A)^2), n))}
|
|
|
CROSSREFS
| A139216(n) = -(-1)^n * a(n).
Sequence in context: A050948 A062590 * A139216 A092078 A067109 A030219
Adjacent sequences: A139212 A139213 A139214 * A139216 A139217 A139218
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Michael Somos, Apr 11 2008
|
| |
|
|
