|
Search: id:A145466
|
|
|
| A145466 |
|
Expansion of q^(1/6) * eta(q) / eta(q^5) in powers of q. |
|
+0
3
|
|
| 1, -1, -1, 0, 0, 2, -1, 0, 0, 0, 3, -2, -2, 0, 0, 4, -3, -2, 0, 0, 7, -5, -3, 0, 0, 10, -6, -4, 0, 0, 15, -10, -7, 0, 0, 20, -13, -8, 0, 0, 28, -19, -13, 0, 0, 38, -25, -16, 0, 0, 52, -34, -23, 0, 0, 68, -44, -28, 0, 0, 91, -60, -40, 0, 0, 118, -76, -48, 0, 0, 153, -100, -66, 0, 0, 196, -127, -82, 0, 0, 252, -164, -107, 0, 0
(list; graph; listen)
|
|
|
OFFSET
|
0,6
|
|
|
FORMULA
|
Expansion of 1 / (G(x) * H(x)) = G(x^5)^2 - x * G(x^5) * H(x^5) - x^2 * H(x^5)^2 in powers of x where G(), H() are the Rogers-Ramanujan functions.
Euler transform of period 5 sequence [ -1, -1, -1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1/(180 t)) = 5^(1/2) / f(t) where q = exp(2 pi i t).
Given g.f. A(x), then B(x) = A(x^3)^2 / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - 5*u*v - u^2*v^2.
Given g.f. A(x), then B(x) = A(x^3)^2 / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v * u^2 * w^2 + 5 * u * w * (u + w) - v^2 * (u^2 + u*w + w^2).
a(5*n + 3) = a(5*n + 4) = 0.
G.f.: 1 / (Product_{k>0} P(5, x^k)) where P(n,x) is nth cyclotomic polynomial.
|
|
EXAMPLE
|
1/q - q^5 - q^11 + 2*q^29 - q^35 + 3*q^59 - 2*q^65 - 2*q^71 + 4*q^89 + ...
|
|
PROGRAM
|
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^5 + A), n))}
|
|
CROSSREFS
|
Convolution inverse of A035959. - A035969(n) = a(5*n + 1). A145467(n) = a(5*n). - A145468(n) = a(5*n+2).
Sequence in context: A037855 A037873 A036869 this_sequence A036868 A130116 A065860
Adjacent sequences: A145463 A145464 A145465 this_sequence A145467 A145468 A145469
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
Michael Somos, Oct 11 2008
|
|
|
Search completed in 0.002 seconds
|
