OFFSET
1,2
COMMENTS
REFERENCES
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). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions
W. Stein, Modular Forms Database.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
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 ((2*k(q)*k'(q))^2 + (k(q)*k'(q))^4) * (K(q) / (pi/2))^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^(4*e) * (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) where a(p) = 2 * Re( (x + i*y)^8 ) and p = x^2 + y^2 with even x. - Michael Somos, Nov 18 2014
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. where i^2 = -1. a(2*n) = 16*a(n). a(4*n + 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 a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^9 (t / i)^9 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 25 2007
EXAMPLE
G.f. = q + 16*q^2 + 256*q^4 - 1054*q^5 + 4096*q^8 + 6561*q^9 - 16864*q^10 - ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^18 (QPochhammer[ -q, q^2]^12 + 4 q / QPochhammer[ -q, q^2]^12), {q, 0, n}]; (* Michael Somos, Apr 12 2013 *)
PROG
(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 */
(PARI) {a(n) = local(A, p, e, x, y, z, a0, a1); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e = A[k, 2]; if( p==2, 16^e, if( p%4 == 3, if( e%2, 0, p^(4*e)), forstep( i=0, sqrtint(p), 2, if( issquare( p - i^2, &y), x = i; break)); a0 = 1; a1 = x = real( (x + I*y)^8 ) * 2; for( i=2, e, y = x*a1 - p^8*a0; a0=a1; a1=y); a1))))) }; /* Michael Somos, Nov 18 2014 */
(Sage) A = CuspForms( Gamma1(4), 9, prec=53).basis(); A[0] + 16*A[1]; # Michael Somos, Apr 12 2013
(Magma) A := Basis( CuspForms( Gamma1(4), 9), 50); A[1] + 16*A[2]; /* Michael Somos, Nov 16 2014 */
CROSSREFS
KEYWORD
sign,mult
AUTHOR
EXTENSIONS
Edited by Michael Somos, Mar 09, 2006
STATUS
approved