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.

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.yu), 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

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

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 */

Cf. A076577, A004017, A045825, A096960.

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

Sequence in context: A045850 A033580 A002408 this_sequence A007259 A101127 A134747

Adjacent sequences: A007328 A007329 A007330 this_sequence A007332 A007333 A007334

easy,nice,nonn,mult

njas, Mira Bernstein (mira(AT)math.berkeley.edu)

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