|
| |
|
|
A164268
|
|
Expansion of f(q^3) * phi(q^3)^3 / (q * f(q^9)^3 * phi(q)) in powers of q where f(), phi() are Ramanujan theta functions.
|
|
3
| |
|
|
1, -2, 4, -1, 0, 4, 1, 0, 0, 1, 0, -8, -1, 0, -8, 0, 0, 4, 1, 0, 16, -2, 0, 16, 0, 0, -4, 2, 0, -32, -3, 0, -32, 1, 0, 8, 4, 0, 56, -4, 0, 56, 1, 0, -16, 4, 0, -96, -6, 0, -92, 1, 0, 24, 5, 0, 160, -8, 0, 152, 1, 0, -40, 8, 0, -252, -10, 0, -240, 2, 0, 64, 11, 0, 392, -14, 0, 368, 4, 0, -96, 14, 0, -600, -19, 0, -560, 4, 0
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| -1,2
|
|
|
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 chi(q^3)^3 / (q * chi(q)) - 2 + 4 * q * chi(q) / chi(q^3)^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^6)^18 * eta(q^9)^3 * eta(q^36)^3 / (eta(q^2)^5 * eta(q^3)^7 * eta(q^12)^7 * eta(q^18)^9) in powers of q.
Euler transform of period 36 sequence [ -2, 3, 5, 1, -2, -8, -2, 1, 2, 3, -2, -3, -2, 3, 5, 1, -2, -2, -2, 1, 5, 3, -2, -3, -2, 3, 2, 1, -2, -8, -2, 1, 5, 3, -2, 0, ...].
a(3*n) = 0 unless n=0.
|
|
|
EXAMPLE
| 1/q - 2 + 4*q - q^2 + 4*q^4 + q^5 + q^8 - 8*q^10 - q^11 - 8*q^13 + ...
|
|
|
PROG
| (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^18 * eta(x^9 + A)^3 * eta(x^36 + A)^3 / (eta(x^2 + A)^5 * eta(x^3 + A)^7 * eta(x^12 + A)^7 * eta(x^18 + A)^9), n))}
|
|
|
CROSSREFS
| Cf. A062244(n) = a(3*n - 1). 4 * A128111(n) = a(3*n + 1).
Sequence in context: A070678 A124091 A067849 * A152433 A094344 A137391
Adjacent sequences: A164265 A164266 A164267 * A164269 A164270 A164271
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Michael Somos, Aug 11 2009
|
| |
|
|
