0,3

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.

Number of representations of n-1 as sum of 8 triangular numbers.

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.

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

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 187.

T. D. Noe, Table of n, a(n) for n=0..1001

B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.

H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares

H. Rosengren, Sums of triangular numbers from the Frobenius determinant

G.f.: q * Product (1+q^(2*k-1))^8*(1+q^(4*k))^8, k=1..inf.

a(n)=sum_{0<d|n, n/d odd} d^3.

Also expansion of Jacobi theta constant theta_2^8/256.

G.f.: Sum_{n>0} n^3*x^n/(1-x^(2*n)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 24 2002

Euler transform of period 2 sequence [8, -8, ...]. - Michael Somos May 31 2005

Expansion of eta(q^2)^16/eta(q)^8 in powers of q. - Michael Somos May 31 2005

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

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_{gamma,2}^2*E_{0,4}=q+8q+28q^2+....

q + 8*q^2 + 28*q^3 + 64*q^4 + 126*q^5 + 224*q^6 + 344*q^7 + 512*q^8 + ...

q*product( (1+q^(2*k-1))^8*(1+q^(4*k))^8, k=1..75);

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, (n/d%2)*d^3)) /* Michael Somos May 31 2005 */

(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 */

(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 */

Cf. A076577, A004017, A045825, A096960.

Cf. A002408(n)=-(-1)^n*a(n).

Sequence in context: A001486 A045850 A033580 this_sequence A002408 A101127 A007259

Adjacent sequences: A007328 A007329 A007330 this_sequence A007332 A007333 A007334

easy,nice,nonn,mult

N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)

Additional comments from Barry Brent (barryb(AT)primenet.com)