|
Search: id:A134756
|
|
|
| A134756 |
|
Coefficients of a q-series of Zagier related to the Dedekind eta function. |
|
+0
1
|
|
| 1, -5, -7, 0, 0, 11, 0, 13, 0, 0, 0, 0, -17, 0, 0, -19, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, -43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 47, 0, 0, 0, 0, 0, 0
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
Obtained by formally "differentiating the Dedekind eta-function half a time".
|
|
REFERENCES
|
D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40(5) (2001), 945-960. See Eq. (20)
|
|
FORMULA
|
a(n) = b(24*n+1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2* p^(e/2) if p == 1, 11 (mod 12), b(p^e) = (1+(-1)^e)/2* (-p)^(e/2) if p == 5, 7 (mod 12).
G.f.: Sum_{k>0} kronecker(12, k) * k * x^((k^2 - 1) / 24).
|
|
EXAMPLE
|
q - 5*q^25 - 7*q^49 + 11*q^121 + 13*q^169 - 17*q^289 - 19*q^361 + ...
|
|
PROGRAM
|
(PARI) {a(n) = if( issquare( 24*n+1, &n), n * kronecker( 12, n), 0)}
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(24*n+1); prod(k = 1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( (p<5) | (e%2), 0, (kronecker( 12, p) * p)^(e\2)))))}
|
|
CROSSREFS
|
sqrt(24*n+1) * A010815(n) = a(n).
Apart from signs, same as A080332, A116916 and A133079. - njas, Nov 11 2007
Sequence in context: A116916 A133079 A080332 this_sequence A011350 A013706 A085679
Adjacent sequences: A134753 A134754 A134755 this_sequence A134757 A134758 A134759
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
Michael Somos, Nov 08 2007
|
|
|
Search completed in 0.002 seconds
|
