|
| |
|
|
A139093
|
|
Expansion of eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3) in powers of q.
|
|
1
| |
|
|
1, 2, -2, -4, 2, 0, -4, 0, 2, 6, 0, -4, 4, 0, 0, 0, 2, 4, -6, -4, 0, 0, -4, 0, 4, 2, 0, -8, 0, 0, 0, 0, 2, 8, -4, 0, 6, 0, -4, 0, 0, 4, 0, -4, 4, 0, 0, 0, 4, 2, -2, -8, 0, 0, -8, 0, 0, 8, 0, -4, 0, 0, 0, 0, 2, 0, -8, -4, 4, 0, 0, 0, 6, 4, 0, -4, 4, 0, 0, 0, 0, 10, -4, -4, 0, 0, -4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, -2, -12, 2, 0, -8, 0
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is in A002479.
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
FORMULA
| Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 4 sequence [ 2, -5, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A112603.
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1))^2 / (1 + x^(2*k)).
a(8*n + 5) = a(8*n + 7) = 0.
A082564(n) = (-1)^n * a(n). A133692(n) = a(2*n). 2 * A125095(n) = a(2*n + 1). A033715(n) = a(4*n). 2 * A112603(n) = a(8*n). -4 * A033761(n) = a(8*n + 3).
|
|
|
EXAMPLE
| 1 + 2*q - 2*q^2 - 4*q^3 + 2*q^4 - 4*q^6 + 2*q^8 + 6*q^9 - 4*q^11 + ...
|
|
|
PROG
| (PARI) {a(n) = if( n<1, n==0, 2 * (-1)^(n\2) * sumdiv(n, d, kronecker(-8, d)))}
(PARI) {a(n) = local(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / eta(x + A)^2 / eta(x^4 + A)^3, n))}
|
|
|
CROSSREFS
| Cf. A002479, A033715, A033761, A082564, A112603, A125095, A133692.
Sequence in context: A033715 A082564 A133692 * A080918 A033758 A033750
Adjacent sequences: A139090 A139091 A139092 * A139094 A139095 A139096
|
|
|
KEYWORD
| sign
|
|
|
AUTHOR
| Michael Somos, Apr 08 2008
|
| |
|
|
