%I #33 Feb 18 2022 20:48:31

%S -1,0,0,4,6,0,0,12,12,0,0,12,16,0,0,24,18,0,0,12,24,0,0,36,24,0,0,16,

%T 24,0,0,36,36,0,0,24,30,0,0,48,24,0,0,12,48,0,0,60,40,0,0,24,24,0,0,

%U 48,48,0,0,36,48,0,0,60,42,0,0,12,48,0,0,84,36,0,0

%N a(n) = 12*H(n) where H() is the Hurwitz class number.

%C Coefficients of q-expansion of Eisenstein series G_{3/2}(tau) multiplied by 12. - _N. J. A. Sloane_, Mar 16 2019

%H Seiichi Manyama, <a href="/A259825/b259825.txt">Table of n, a(n) for n = 0..10000</a>

%H K. Bringmann and J. Lovejoy, <a href="http://arxiv.org/abs/0712.0631">Overpartitions and class numbers of binary quadratic forms</a>, arXiv:0712.0631 [math.NT], 2007. See page 5, equation (1.12).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hurwitz_class_number">Hurwitz class number</a>.

%H D. Zagier, <a href="https://people.mpim-bonn.mpg.de/zagier/files/tex/UtrechtLectures/UtBook.pdf">Modular Forms of One Variable</a>, Notes based on a course given in Utrecht, 1991. See page 50.

%F a(n) = 12 * A058305(n) / A058306(n). a(4*n + 1) = a(4*n + 2) = 0. a(3*n + 4) = 6 * A259827(n).

%F a(4*n + 3) = 4 * A130695(n). a(8*n + 3) = A005886(n) = 2 * A005869(n) = 4 * A008443(n). a(12*n + 7) = 12 * A259655(n).

%F a(16*n + 4) = 6 * A045834(n) = 3 * A005876(n). a(16*n + 8) = 12 * A045828(n) = 6 * A005884(n) = 3 * A005877(n).

%F a(24*n + 3) = 4 * A213627(n). a(24*n + 7) = 12 * A185220(n). a(24*n + 11) = 12 * A213617(n). a(24*n + 19) = 12 * A181648(n). a(24*n + 23) = 12 * A188569(n+1).

%F a(32*n + 4) = 6 * A213022(n). a(32*n + 8) = 12 * A213625(n). a(32*n + 12) = 16 * A008443(n) = 8 * A005869(n) = 4 * A005886(n) = 2 * A005878(n). a(32*n + 20) = 24 * A045831(n) = 6 * A004024(n). a(32*n + 24) = 24 * A213624(n).

%F G.f.: -2 * (Sum_{k in Z} (-1)^k * x^(k*k + k) / (1 + (-x)^k)^2) / (Sum_{k in Z} x^k^2) - 2 * (Sum_{k in Z} (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2) / (Sum_{k in Z} (-x)^k^2).

%F a(n) >= 0 if n > 0. - _Michael Somos_, Feb 04 2022

%e G.f. = -1 + 4*x^3 + 6*x^4 + 12*x^7 + 12*x^8 + 12*x^11 + 16*x^12 + 24*x^15 + ...

%t terms = 100; gf[m_] := With[{r = Range[-m, m]}, -2 Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, r}]/EllipticTheta[3, 0, x] - 2 Sum[(-1)^k*x^(k^2 + 2 k)/(1 + x^(2 k))^2, {k, r}]/EllipticTheta[3, 0, -x]]; gf[terms // Sqrt // Ceiling] + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Apr 02 2017 *)

%t a[ n_] := If[ n<1, -Boole[n==0], With[{m = Floor[(-1 + Sqrt[1 + 4*n])/2]}, -2*SeriesCoefficient[ Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, -m-1,m}] / EllipticTheta[3, 0, x] + Sum[(-1)^k*x^(k^2 + 2*k)/(1 + x^(2*k))^2, {k, -m-2,m}]/ EllipticTheta[3, 0, -x], {x, 0, n}]]]; (* _Michael Somos_, Feb 04 2022 *)

%o (PARI) {a(n) = 12 * qfbhclassno(n)};

%o (PARI) {a(n) = my(D, f); 12 * if( n<1, (n==0)/-12, [D, f] = core(-n, 1); if( D%4>1 && !(f%2), D*=4; f/=2); if( D%4<2, qfbclassno(D) / max(1, D+6), 0) * sumdiv(f, d, moebius(d) * kronecker(D, d) * sigma(f/d)))};

%Y Cf. A004024, A005869, A005876, A005877, A005878, A005884, A005886, A008443.

%Y Cf. A045828, A045831, A045834, A058305, A058306, A130695, A181648, A185220.

%Y Cf. A188569, A213022, A213617, A213624, A213625, A213627, A259827.

%Y See also A306934-A306937.

%K sign

%O 0,4

%A _Michael Somos_, Jul 05 2015