A134756 Coefficients of a q-series of Zagier related to the Dedekind eta function.

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

Offset

0,2

Comments

Obtained by formally "differentiating the Dedekind eta-function half a time".

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, vol.40, pp.945-960 (2001). See Eq. (20).

Formula

a(n) = b(24*n + 1) where b() 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).

a(n) = sqrt(24*n + 1) * A010815(n).

Example

G.f. = 1 - 5*x - 7*x^2 + 11*x^5 + 13*x^7 - 17*x^12 - 19*x^15 + 23*x^22 + ...
G.f. = q - 5*q^25 - 7*q^49 + 11*q^121 + 13*q^169 - 17*q^289 - 19*q^361 + ...

Mathematica

a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ @ m, m KroneckerSymbol[ 12, m], 0]]; (* Michael Somos, Oct 15 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (If[ # < 5, 0, (1 - Mod[#2, 2]) (# KroneckerSymbol[ 12, #])^(#2/2)] & @@@ FactorInteger[ 24 n + 1])]; (* Michael Somos, Oct 15 2015 *)
s = QPochhammer[q] + O[q]^100; A010815 = CoefficientList[s, q]; nn = Range[0, Length[A010815]-1]; A134756 = Sqrt[24*nn+1]*A010815 (* Jean-François Alcover, Dec 01 2015 *)

Prog

(PARI) {a(n) = if( issquare( 24*n+1, &n), n * kronecker( 12, n), 0)};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(24*n+1); prod(k = 1, matsize(A)[1], [p, e] = A[k, ]; if( (p<5) || (e%2), 0, (kronecker( 12, p) * p)^(e\2))))};

Crossrefs

Cf. A010815.

Apart from signs, same as A080332, A116916 and A133079. - N. J. A. Sloane, Nov 11 2007

Sequence in context: A133079 A116916 A080332 * A178902 A176713 A293506

Adjacent sequences: A134753 A134754 A134755 * A134757 A134758 A134759

Keyword

sign

Author

Michael Somos, Nov 08 2007

Status

approved