A006352 - OEIS

%I M5145 #74 Aug 22 2022 11:54:41

%S 1,-24,-72,-96,-168,-144,-288,-192,-360,-312,-432,-288,-672,-336,-576,

%T -576,-744,-432,-936,-480,-1008,-768,-864,-576,-1440,-744,-1008,-960,

%U -1344,-720,-1728,-768,-1512,-1152,-1296,-1152,-2184,-912,-1440,-1344,-2160,-1008,-2304,-1056,-2016,-1872,-1728

%N Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).

%C Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

%C The series Q(q), R(q) are modular forms, but P(q) is not. - _Michael Somos_, May 18 2017

%D R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.

%D N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see pp. 111 and 113.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 19, Eq. (17).

%H T. D. Noe, <a href="/A006352/b006352.txt">Table of n, a(n) for n = 0..1000</a>

%H F. Beukers, <a href="http://dx.doi.org/10.1016/0022-314X(87)90025-4">Another congruence for the Apéry numbers</a>, J. Number Theory 25 (1987), no. 2, 201-210.

%H J. H. Bruinier and K. Ono, <a href="http://www.aimath.org/news/partition/brunier-ono.pdf">Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms</a>

%H Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, <a href="http://unimodular.net/archive/RamEisenstein.pdf">Ramanujan's Eisenstein series and powers of Dedekind's eta-function</a>, Journal of the London Mathematical Society 75.1 (2007): 225-242. See P(q).

%H M. Kaneko and D. Zagier, <a href="http://www2.math.kyushu-u.ac.jp/~mkaneko/papers/atkin.pdf">Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials</a>, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998

%H H. Ochiai, <a href="http://arXiv.org/abs/math-ph/9909023">Counting functions for branched covers of elliptic curves and quasi-modular forms</a>, arXiv:math-ph/9909023, 1999.

%H Varin, V. P.  <a href="https://doi.org/10.1134/S0965542517020154">Special solutions to Chazy equation</a>  Comput. Math. Math. Phys. 57, No. 2, 211-235 (2017), eq (75)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series.</a>

%H <a href="/index/Ed#Eisen">Index entries for sequences related to Eisenstein series</a>

%F a(n) = -24*sigma(n) = -24*A000203(n), for n>0.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 4*u2^2 + 9*u3^2 + 36*u6^2 - 8*u1*u2 + 6*u1*u3 + 24*u2*u6 - 72*u3*u6. - _Michael Somos_, May 29 2005

%F G.f.: 1 - 24*sum(k>=1, k*x^k/(1 - x^k)).

%F G.f.: 1 + 24 *x*deriv(eta(x))/eta(x) where eta(x) = prod(n>=1, 1-x^n); (cf. A000203). - _Joerg Arndt_, Sep 28 2012

%F G.f.: 1 - 24*x/(1-x) + 48*x^2/(Q(0) -  2*x^2 + 2*x), where Q(k)= (2*x^(k+2) - x - 1)*k - 1 - 2*x + 3*x^(k+2) - x*(k+1)*(k+3)*(1-x^(k+2))^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 16 2013

%F G.f.: q*Delta'/Delta where Delta is the generating function of Ramanujan's tau function (A000594). - _Seiichi Manyama_, Jul 15 2017

%e G.f. = 1 - 24*x - 72*x^2 - 96*x^3 - 168*x^4 - 144*x^5 - 288*x^6 + ...

%p E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(2);

%t a[n_] := -24*DivisorSigma[1, n]; a[0] = 1; Table[a[n], {n, 0, 46}] (* _Jean-François Alcover_, Dec 12 2012 *)

%t a[ n_] := If[ n < 1, Boole[n == 0], -24 DivisorSigma[ 1, n]]; (* _Michael Somos_, Apr 08 2015 *)

%o (PARI) {a(n) = if( n<1, n==0, -24 * sigma(n))}; /* _Michael Somos_, Apr 09 2003 */

%o (Python)

%o from sympy import divisor_sigma

%o def a(n): return 1 if n == 0 else -24 * divisor_sigma(n)

%o [a(n) for n in range(51)]  # _Indranil Ghosh_, Jul 15 2017

%Y Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).

%Y Cf. A000594 (Delta), A076835, A145155 (Delta').

%K sign,easy,nice

%O 0,2

%A _N. J. A. Sloane_