%I #39 Aug 06 2024 04:33:34

%S 1,480,61920,1050240,7926240,37500480,135480960,395301120,1014559200,

%T 2296875360,4837561920,9353842560,17342613120,30119288640,50993844480,

%U 82051050240,129863578080,196962563520

%N a(0) = 1, a(n) = 480*sigma_7(n).

%C Eisenstein series E_8(q) (alternate convention E_4(q)); theta series of direct sum of 2 copies of E_8 lattice.

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.

%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 p. 111.

%D S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.

%D S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.

%H Seiichi Manyama, <a href="/A008410/b008410.txt">Table of n, a(n) for n = 0..10000</a>

%H H. D. Nguyen, D. Taggart, <a href="https://citeseerx.ist.psu.edu/pdf/8f2f36f22878c984775ed04368b8893879b99458">Mining the OEIS: Ten Experimental Conjectures</a>, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014

%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 Equivalently, g.f. = (theta2^16+theta3^16+theta4^16)/2.

%F G.f. Sum{k>=0} a(k)q^(2k) = (theta2^16+theta3^16+theta4^16)/2.

%F Expansion of ((eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8)^2 in powers of q. - _Michael Somos_, Dec 30 2008

%F G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^8 * f(t) where q = exp(2 Pi i t). - _Michael Somos_, Dec 30 2008

%F a(n) = 480*A013955(n). - _R. J. Mathar_, Oct 10 2012

%e G.f. = 1 + 480*q + 61920*q^2 + 1050240*q^3 + 7926240*q^4 + 37500480*q^5 + ...

%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(8);

%t a[ n_] := If[ n < 1, Boole[n == 0], 480 DivisorSigma[ 7, n]]; (* _Michael Somos_, Jun 04 2013 *)

%t nmax = 60; CoefficientList[Series[(Product[(1-x^k)^8 / (1+x^k)^8, {k, 1, nmax}] + 256 * x * Product[(1+x^k)^16 *(1-x^k)^8, {k, 1, nmax}])^2, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 02 2017 *)

%o (PARI) {a(n) = if( n<1, n==0, 480 * sigma(n, 7))};

%o (PARI) {a(n) = local(A, e1, e2, e4); if( n<0, 0, n*=2; A = x * O(x^n); e1 = eta(x + A)^16; e2 = eta(x^2 + A)^16; e4 = eta(x^4 + A)^16; polcoeff( (e1*e2^3 + 256*x^2 * e4*(e2^3 + e1^2*e4)) / (e1*e2*e4), n))}; /* _Michael Somos_, Jun 29 2005 */

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8)^2, n))}; /* _Michael Somos_, Dec 30 2008 */

%o (Sage) ModularForms( Gamma1(1), 8, prec=33).0; # _Michael Somos_, Jun 04 2013

%o (Magma) Basis( ModularForms( Gamma1(1), 8), 33) [1]; /* _Michael Somos_, May 27 2014 */

%Y Cf. A013973.

%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 Convolution square of A004009.

%K nonn

%O 0,2

%A _N. J. A. Sloane_