|
Search: id:A121455
|
|
|
| A121455 |
|
Expansion of q*(phi(-q)psi(q^4))^2 in powers of q where phi(),psi() are Ramanujan theta functions. |
|
+0
3
|
|
| 1, -4, 4, 0, 6, -16, 8, 0, 13, -24, 12, 0, 14, -32, 24, 0, 18, -52, 20, 0, 32, -48, 24, 0, 31, -56, 40, 0, 30, -96, 32, 0, 48, -72, 48, 0, 38, -80, 56, 0, 42, -128, 44, 0, 78, -96, 48, 0, 57, -124, 72, 0, 54, -160, 72, 0, 80, -120, 60, 0, 62, -128, 104, 0, 84, -192, 68, 0, 96, -192, 72, 0, 74, -152
(list; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
FORMULA
|
Expansion of (eta(q)eta(q^8))^4/(eta(q^2)eta(q^4))^2 in powers of q.
Euler transform of period 8 sequence [ -4, -2, -4, 0, -4, -2, -4, -4, ...].
Multiplicative with a(2)=-4, a(2^e)=0 if e>1, a(p^e)=(p^(e+1)-1)/(p-1) if p>2.
a(4n)=0. a(4n+2)=-4*sigma(2n+1). a(2n+1)=sigma(2n+1).
G.f. is Fourier series of a weight 2 level 8 cusp form. f(-1/ (8 t)) = -8 t^2 f(t) where q = exp(2 pi i t) .
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w)= ( v* (v+2*w)* (u+2*v))^2 -16* (u*w)^3 -u*v*w* (u +2*v +4*w) *(u^2 +16*v^2 +16*w^2 +10*u*v +28*u*w +40*v*w) .
|
|
EXAMPLE
|
q - 4*q^2 + 4*q^3 + 6*q^5 - 16*q^6 + 8*q^7 + 13*q^9 - 24*q^10 + 12*q^11 + ...
|
|
PROGRAM
|
(PARI) {a(n)=if(n<1, 0, if(n%2, sigma(n), if(n/2%2, -4*sigma(n/2), 0)))}
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^8+A))^4/(eta(x^2+A)*eta(x^4+A))^2, n))}
|
|
CROSSREFS
|
Sequence in context: A069523 A021231 A016705 this_sequence A133657 A129507 A021698
Adjacent sequences: A121452 A121453 A121454 this_sequence A121456 A121457 A121458
|
|
KEYWORD
|
sign,mult
|
|
AUTHOR
|
Michael Somos, Jul 30 2006, May 28 2007
|
|
|
Search completed in 0.002 seconds
|
