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A008410 a(0) = 1, a(n) = 480*sigma_7(n).

%I A008410 #40 Feb 16 2025 08:32:32
%S A008410 1,480,61920,1050240,7926240,37500480,135480960,395301120,1014559200,
%T A008410 2296875360,4837561920,9353842560,17342613120,30119288640,50993844480,
%U A008410 82051050240,129863578080,196962563520
%N A008410 a(0) = 1, a(n) = 480*sigma_7(n).
%C A008410 Eisenstein series E_8(q) (alternate convention E_4(q)); theta series of direct sum of 2 copies of E_8 lattice.
%D A008410 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
%D A008410 R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
%D A008410 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
%D A008410 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 A008410 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 A008410 Seiichi Manyama, <a href="/A008410/b008410.txt">Table of n, a(n) for n = 0..10000</a>
%H A008410 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 A008410 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series.</a>
%H A008410 <a href="/index/Ed#Eisen">Index entries for sequences related to Eisenstein series</a>
%F A008410 Equivalently, g.f. = (theta2^16+theta3^16+theta4^16)/2.
%F A008410 G.f. Sum{k>=0} a(k)q^(2k) = (theta2^16+theta3^16+theta4^16)/2.
%F A008410 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 A008410 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 A008410 a(n) = 480*A013955(n). - _R. J. Mathar_, Oct 10 2012
%e A008410 G.f. = 1 + 480*q + 61920*q^2 + 1050240*q^3 + 7926240*q^4 + 37500480*q^5 + ...
%p A008410 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 A008410 a[ n_] := If[ n < 1, Boole[n == 0], 480 DivisorSigma[ 7, n]]; (* _Michael Somos_, Jun 04 2013 *)
%t A008410 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 A008410 (PARI) {a(n) = if( n<1, n==0, 480 * sigma(n, 7))};
%o A008410 (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 A008410 (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 A008410 (Sage) ModularForms( Gamma1(1), 8, prec=33).0; # _Michael Somos_, Jun 04 2013
%o A008410 (Magma) Basis( ModularForms( Gamma1(1), 8), 33) [1]; /* _Michael Somos_, May 27 2014 */
%Y A008410 Cf. A013973.
%Y A008410 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 A008410 Convolution square of A004009.
%K A008410 nonn
%O A008410 0,2
%A A008410 _N. J. A. Sloane_