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%I #18 Sep 07 2018 01:41:26
%S 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,
%T 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,
%U 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
%N Coefficients of a q-series of Zagier related to the Dedekind eta function.
%C Obtained by formally "differentiating the Dedekind eta-function half a time".
%H G. C. Greubel, <a href="/A134756/b134756.txt">Table of n, a(n) for n = 0..1000</a>
%H Don Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1016/S0040-9383(00)00005-7/fulltext.pdf">Vassiliev invariants and a strange identity related to the Dedekind eta-function</a>, Topology, vol.40, pp.945-960 (2001). See Eq. (20).
%F 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).
%F G.f.: Sum_{k>0} Kronecker(12, k) * k * x^((k^2 - 1) / 24).
%F a(n) = sqrt(24*n + 1) * A010815(n).
%e G.f. = 1 - 5*x - 7*x^2 + 11*x^5 + 13*x^7 - 17*x^12 - 19*x^15 + 23*x^22 + ...
%e G.f. = q - 5*q^25 - 7*q^49 + 11*q^121 + 13*q^169 - 17*q^289 - 19*q^361 + ...
%t a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ @ m, m KroneckerSymbol[ 12, m], 0]]; (* _Michael Somos_, Oct 15 2015 *)
%t 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 *)
%t 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 *)
%o (PARI) {a(n) = if( issquare( 24*n+1, &n), n * kronecker( 12, n), 0)};
%o (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))))};
%Y Cf. A010815.
%Y Apart from signs, same as A080332, A116916 and A133079. - _N. J. A. Sloane_, Nov 11 2007
%K sign
%O 0,2
%A _Michael Somos_, Nov 08 2007
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