1,2

J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Pure and Appl. Math. 38 (1907), 289-351 (see p. 304).

W. Stein, Modular Forms Database.

Expansion of newform of degree 1, level 4, weight 9 and nontrivial character in powers of q. - Michael Somos Mar 09 2006

Expansion of Jacobi ((2kk')^2+(kk')^4)(2K/pi)^9/64 in powers of q. - Michael Somos Mar 09 2006

Expansion of F(phi(q)^4,q*psi(q^2)^4) in powers of q where F(u,v)=sqrt(u)*v*(u-16*v)*(u^2+4*u*v-64*v^2), and phi(),psi() are Ramanujan theta functions. - Michael Somos Mar 09 2006

a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = p^(4e)*(1+(-1)^e)/2 if p == 3 (mod 4), a(p^e) = a(p)*a(p^(e-1)) - p^8*a(p^(e-2)) if p == 1 (mod 4) . - Michael Somos Mar 09 2006

G.f.: (t''''*t -28*t'''*t' +35*t''^2)/2 where t=phi(q) and f' := q*df/dq . - Michael Somos Mar 09 2006

G.f.: ( Sum_{j,k} (j+i*k)^8* x^(j^2+k^2) )/4 . a(4n+3)=0.

Expansion of q* f(-q^2)^18* (chi(q)^12 +4*q/ chi(q)^12) in powers of q where f(), chi() are Ramanujan theta functions. - Michael Somos Jul 25 2007

G.f. is Fourier series of a weight 9 level 4 cusp form. f(-1/ (4 t)) = i (-2 t)^9 f(t) where q = exp(2 pi i t). - Michael Somos Jul 25 2007

q + 16*q^2 + 256*q^4 - 1054*q^5 + 4096*q^8 + 6561*q^9 - 16864*q^10 - ...

(PARI) {a(n)=local(m); if(n<1, 0, m=sqrtint(n); polcoeff( sum(j=-m, m, sum(k=-m, m, (j+I*k)^8* x^(j^2+k^2), x*O(x^n)))/4, n))} /* Michael Somos Mar 09 2006 */

(PARI) {a(n)= local(A, B); if(n<1, 0, n--; A= x*O(x^n); B= (eta(x^2+A)^2/ eta(x+A)/ eta(x^4+A))^12; polcoeff( eta(x^2+A)^18* (B +4*x/B), n))} /* Michael Somos Jul 25 2007 */

Adjacent sequences: A002604 A002605 A002606 this_sequence A002608 A002609 A002610

Sequence in context: A059681 A135925 A123935 this_sequence A111979 A050467 A008835

sign,mult

njas

Edited by Michael Somos, Mar 09, 2006