A306934 - OEIS

%I #18 Feb 25 2024 12:41:25

%S 1,-10,0,0,-70,-48,0,0,-120,-250,0,0,-240,-240,0,0,-550,-480,0,0,-528,

%T -480,0,0,-720,-1210,0,0,-960,-720,0,0,-1080,-1440,0,0,-1750,-1200,0,

%U 0,-1680,-1920,0,0,-1680,-1488,0,0,-2160,-3370,0,0,-2640,-1680,0,0,-2400,-3360,0,0,-2880,-2640

%N Coefficients of q-expansion of Eisenstein series G_{5/2}(tau) multiplied by 120.

%H H. Cohen, <a href="https://doi.org/10.1007/BF01436180">Sums involving the values at negative integers of L-functions of quadratic characters</a>, Math. Ann. 217 (1975), no. 3, 271-285.

%H X. Wang and D. Pei, <a href="https://doi.org/10.1007/978-3-642-29302-3">Modular Forms with Integral and Half-Integral Weights</a>, Science Press Beijing, Springer Berlin Heidelberg, 2012. x+432 pp.

%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 (erroneously gives a(5) = -72).

%o (Sage)

%o def a(n):

%o     if n==0: return 1

%o     if (n%4) not in [0,1]: return 0

%o     D = Integer(n).squarefree_part()

%o     f = Integer(sqrt(n/D))

%o     if (D%4) not in [0,1]: D, f = 4*D, f//2

%o     X = kronecker_character(D)

%o     s = sum([moebius(d)*X(d)*d*sigma(f/d, 3) for d in f.divisors()])

%o     return round((120*X.lfunction(100)(-1)*s).real()) # _Robin Visser_, Feb 24 2024

%Y Cf. A259825, A306935, A306936, A306937.

%K sign

%O 0,2

%A _N. J. A. Sloane_, Mar 16 2019

%E Corrected and more terms from _Robin Visser_, Feb 24 2024