|
Search: id:A130539
|
|
|
| A130539 |
|
Expansion of a(q)*b(q)*c(q)/(3*q^(1/3)) in powers of q where a(),b(),c() are cubic AGM functions. |
|
+0
1
|
|
| 1, 4, -13, 0, -1, 16, 11, 0, 25, -52, -46, 0, 47, 0, -22, 0, 120, -4, 0, 0, -121, 64, -109, 0, -97, 44, 131, 0, 0, 0, 13, 0, 167, 100, -37, 0, -214, -208, 0, 0, 121, -184, 146, 0, -143, 0, 251, 0, 0, 188, 59, 0, -118, 0, 299, 0, -168, -88, -325, 0, -313, 0, 0, 0, 143, 480, -277, 0, 0, -16, -253, 0
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
Denoted by g_3(q) in Cynk and Hulek in Remark 3.4 on page 12
|
|
LINKS
|
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds
W. Stein, Modular Forms Database.
|
|
FORMULA
|
Expansion of q^(-1/3)*( eta(q)^5 *eta(q^3) +9*eta(q)^2 *eta(q^3) *eta(q^9)^3 ) in powers of q.
G.f. is Fourier series of a weight 3 level 27 cusp form. f(-1/ (27 t)) = (-3^9)^(1/2) t^3 f(t) where q = exp(2 pi i t) .
a(n)=b(3n+1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2*p^e if p == 2 (mod 3), b(p^e) = b(p)*b(p^(e-1)) -p^2*b(p^(e-2)) if p == 1 (mod 3) where b(p) = x^2-2p, 4p = x^2+3y^2, |x|<|y|, and x == 2 (mod 3).
G.f.: Sum_{k>=0} a(k)*x^(3*k+1) = (1/2)* Sum_{u,v} (u*u -7*v*v)* x^(u*u +u*v +7*v*v). - Michael Somos Jun 14 2007
|
|
EXAMPLE
|
q + 4*q^4 - 13*q^7 - q^13 + 16*q^16 + 11*q^19 + 25*q^25 - 52*q^28 - ...
|
|
PROGRAM
|
(PARI) {a(n)= local(A, p, e, x, y, a0, a1); n=3*n+1; if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 0, if(p%3==2, if(e%2, 0, p^e), for(x=1, sqrtint(4*p\27), if(issquare(4*p -27*x^2, &y), break)); y = y^2-p*2; a0=1; a1=y; for(i=2, e, x=y*a1 -p^2*a0; a0=a1; a1=x); a1)))))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( eta(x+A)^2 *eta(x^3+A) *(eta(x+A)^3 +9* x* eta(x^9+A)^3), n))}
|
|
CROSSREFS
|
Sequence in context: A104129 A009516 A024248 this_sequence A130650 A051432 A046737
Adjacent sequences: A130536 A130537 A130538 this_sequence A130540 A130541 A130542
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
Michael Somos, Jun 03 2007
|
|
|
Search completed in 0.002 seconds
|
