%I A007331 M4503
%S A007331 0,1,8,28,64,126,224,344,512,757,1008,1332,1792,2198,2752,3528,4096,4914,
%T A007331 6056,6860,8064,9632,10656,12168,14336,15751,17584,20440,22016,24390,
%U A007331 28224,29792,32768,37296,39312,43344,48448,50654,54880,61544,64512
%N A007331 Fourier coefficients of E_{\infty,4}.
%C A007331 E_{\infty,4} is the unique normalized weight-4 modular form for \Gamma_0(2) 
               with simple zeros at i*\infty. Since this has level 2, it is not 
               a cusp form, in contrast to A002408.
%C A007331 Number of representations of n-1 as sum of 8 triangular numbers.
%C A007331 Multiplicative with a(2^e) = 2^(3e), a(p^e) = (p^(3(e+1))-1)/(p^3-1). 
               Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) Jun 13, 2005.
%D A007331 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007331 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 187.
%H A007331 T. D. Noe, <a href="b007331.txt">Table of n, a(n) for n=0..1001</a>
%H A007331 B. Brent, <a href="http://www.expmath.org/expmath/volumes/7/7.html">Quadratic 
               Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.</
               a>
%H A007331 H. H. Chan and C. Krattenthaler, <a href="http://arXiv.org/abs/math.NT/
               0407061">Recent progress in the study of representations of integers 
               as sums of squares</a>
%H A007331 H. Rosengren, <a href="http://arXiv.org/abs/math.NT/0504272">Sums of 
               triangular numbers from the Frobenius determinant</a>
%F A007331 G.f.: q * Product (1+q^(2*k-1))^8*(1+q^(4*k))^8, k=1..inf.
%F A007331 a(n)=sum_{0<d|n, n/d odd} d^3.
%F A007331 Also expansion of Jacobi theta constant theta_2^8/256.
%F A007331 G.f.: Sum_{n>0} n^3*x^n/(1-x^(2*n)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Oct 24 2002
%F A007331 Euler transform of period 2 sequence [8, -8, ...]. - Michael Somos May 
               31 2005
%F A007331 Expansion of eta(q^2)^16/eta(q)^8 in powers of q. - Michael Somos May 
               31 2005
%F A007331 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= v^3 -u^2*w 
               +16*u*v*w -32*v^2*w +256*v*w^2 . - Michael Somos May 31 2005
%F A007331 G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 16^(-1) 
               (t / i)^4 g(t) where g() is g.f. for A035016. - Michael Somos Jan 
               11 2009
%e A007331 E_{gamma,2}^2*E_{0,4}=q+8q+28q^2+....
%e A007331 q + 8*q^2 + 28*q^3 + 64*q^4 + 126*q^5 + 224*q^6 + 344*q^7 + 512*q^8 + 
               ...
%p A007331 q*product( (1+q^(2*k-1))^8*(1+q^(4*k))^8, k=1..75);
%o A007331 (PARI) a(n)=if(n<1, 0, sumdiv(n,d,(n/d%2)*d^3)) /* Michael Somos May 
               31 2005 */
%o A007331 (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x^n*O(x); polcoeff( (eta(x^2+A)^2/
               eta(x+A))^8, n))} /* Michael Somos May 31 2005 */
%o A007331 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x^n * O(x); polcoeff( (eta(x^2 
               + A)^2 / eta(x + A))^8, n))} /* Michael Somos May 31 2005 */
%Y A007331 Cf. A076577, A004017, A045825, A096960.
%Y A007331 Cf. A002408(n)=-(-1)^n*a(n).
%Y A007331 Sequence in context: A001486 A045850 A033580 this_sequence A002408 A101127 
               A007259
%Y A007331 Adjacent sequences: A007328 A007329 A007330 this_sequence A007332 A007333 
               A007334
%K A007331 easy,nice,nonn,mult
%O A007331 0,3
%A A007331 N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)
%E A007331 Additional comments from Barry Brent (barryb(AT)primenet.com)