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Glaisher's chi_8(n).
(Formerly M4994 N2150)
4

%I M4994 N2150 #57 Sep 08 2022 08:44:31

%S 1,16,0,256,-1054,0,0,4096,6561,-16864,0,0,-478,0,0,65536,-63358,

%T 104976,0,-269824,0,0,0,0,720291,-7648,0,0,-1407838,0,0,1048576,0,

%U -1013728,0,1679616,925922,0,0,-4317184,3577922,0,0,0,-6915294,0,0,0,5764801,11524656,0,-122368

%N Glaisher's chi_8(n).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D 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]

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Seiichi Manyama, <a href="/A002607/b002607.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from G. C. Greubel)

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H W. Stein, <a href="http://wstein.org:8080/mfd/space.html?space=[4,9,[1]]&amp;search=4">Modular Forms Database</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>

%F Expansion of newform of degree 1, level 4, weight 9 and nontrivial character in powers of q. - _Michael Somos_, Mar 09 2006

%F 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

%F 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

%F 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

%F 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

%F 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.

%F 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

%F 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

%e G.f. = q + 16*q^2 + 256*q^4 - 1054*q^5 + 4096*q^8 + 6561*q^9 - 16864*q^10 - ...

%t 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 *)

%o (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 */

%o (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 */

%o (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 */

%o (Sage) A = CuspForms( Gamma1(4), 9, prec=53).basis(); A[0] + 16*A[1]; # _Michael Somos_, Apr 12 2013

%o (Magma) A := Basis( CuspForms( Gamma1(4), 9), 50); A[1] + 16*A[2]; /* _Michael Somos_, Nov 16 2014 */

%Y Cf. A030212, A247067.

%K sign,mult

%O 1,2

%A _N. J. A. Sloane_

%E Edited by _Michael Somos_, Mar 09, 2006