|
Search: id:A122857
|
|
|
| A122857 |
|
Expansion of (phi(q)^2+phi(q^3)^2)/2 in powers of q where phi() is a Ramanujan theta function. |
|
+0
2
|
|
| 1, 2, 2, 2, 2, 4, 2, 0, 2, 2, 4, 0, 2, 4, 0, 4, 2, 4, 2, 0, 4, 0, 0, 0, 2, 6, 4, 2, 0, 4, 4, 0, 2, 0, 4, 0, 2, 4, 0, 4, 4, 4, 0, 0, 0, 4, 0, 0, 2, 2, 6, 4, 4, 4, 2, 0, 0, 0, 4, 0, 4, 4, 0, 0, 2, 8, 0, 0, 4, 0, 0, 0, 2, 4, 4, 6, 0, 0, 4, 0, 4, 2, 4, 0, 0, 8, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0, 2, 4, 2, 0, 6, 4, 4, 0, 4
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
FORMULA
|
Expansion of eta(q^2)^3*eta(q^3)^2*eta(q^6)/(eta(q)^2*eta(q^4)*eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 2, -1, 0, 0, 2, -4, 2, 0, 0, -1, 2, -2, ...].
Moebius transform is period 12 sequence [ 2, 0, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, ...].
a(12n+7)=a(12n+11)=0.
a(n) = 2 * b(n) where b(n) is multiplicative and b(2^e) = b(3^e) = 1, b(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) == (1-(-1)^e)/2 if p == 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A125061.
|
|
PROGRAM
|
(PARI) {a(n)=if(n<1, n==0, 2*sumdiv(n, d, kronecker(-36, d)))}
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p<5, 1, if( p%12<6, e+1, !(e%2))))))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)), n))}
|
|
CROSSREFS
|
A035154(n)=a(n)/2 if n>0. A008441(n)=a(4n+1)/2. A035181(2n+1)=a(2n+1)/2. A113446(3n+1)=A002654(3n+1)=a(3n+1)/2.
Sequence in context: A086668 A092904 A062816 this_sequence A132003 A109810 A122066
Adjacent sequences: A122854 A122855 A122856 this_sequence A122858 A122859 A122860
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Michael Somos, Sep 14 2006
|
|
|
Search completed in 0.002 seconds
|
