A002607 Glaisher's chi_8(n). (Formerly M4994 N2150)

1, 16, 0, 256, -1054, 0, 0, 4096, 6561, -16864, 0, 0, -478, 0, 0, 65536, -63358, 104976, 0, -269824, 0, 0, 0, 0, 720291, -7648, 0, 0, -1407838, 0, 0, 1048576, 0, -1013728, 0, 1679616, 925922, 0, 0, -4317184, 3577922, 0, 0, 0, -6915294, 0, 0, 0, 5764801, 11524656, 0, -122368

Offset

1,2

Comments

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

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

Index entries for sequences mentioned by Glaisher

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

Cf. A030212, A247067.

Sequence in context: A188784 A123935 A169764 * A221404 A221726 A221729

Adjacent sequences: A002604 A002605 A002606 * A002608 A002609 A002610

Keyword

sign,mult

Author

N. J. A. Sloane

Extensions

Edited by Michael Somos, Mar 09, 2006

Status

approved