Search: id:A008410
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%I A008410
%S A008410 1,480,61920,1050240,7926240,37500480,135480960,395301120,
%T A008410 1014559200,2296875360,4837561920,9353842560,17342613120,
%U A008410 30119288640,50993844480,82051050240,129863578080,196962563520
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EisensteinSeries.html">Link to a section of The World of Mathematics.</
               a>
%H A008410 <a href="Sindx_Ed.html#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
%e A008410 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);
%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 */
%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.
%Y A008410 Sequence in context: A035314 A022047 A107511 this_sequence A020286 A064909 
               A051980
%Y A008410 Adjacent sequences: A008407 A008408 A008409 this_sequence A008411 A008412 
               A008413
%K A008410 nonn
%O A008410 0,2
%A A008410 N. J. A. Sloane (njas(AT)research.att.com).

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