A002408 Expansion of 8-dimensional cusp form.

0, 1, -8, 28, -64, 126, -224, 344, -512, 757, -1008, 1332, -1792, 2198, -2752, 3528, -4096, 4914, -6056, 6860, -8064, 9632, -10656, 12168, -14336, 15751, -17584, 20440, -22016, 24390, -28224, 29792, -32768, 37296, -39312, 43344, -48448, 50654, -54880, 61544, -64512, 68922

Offset: 0

Author

N. J. A. Sloane and Mira Bernstein

Keywords

• sign
• nice
• easy
• mult

Comments

Essentially the same as A007331.

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

"For Gamma, it is known that any modular form is a weighted homogeneous polynomial in Theta_Z, which has weight 1/2, and the modular form delta_8(t) := e^(Pi i tau) Product_{m=1..oo} ((1 - e^(Pi i m tau)) (1 + e^(2 Pi i m tau)))^8 = e^(Pi i tau) - 8 e^(2 Pi i tau) + 28 e^(3 Pi i tau) - 64 e^(4 Pi i tau) + 126 e^(5 Pi i tau) ... of weight 4." [Elkies, p. 1242]

Examples

			G.f. = q - 8*q^2 + 28*q^3 - 64*q^4 + 126*q^5 - 224*q^6 + 344*q^7 ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 187.
• Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and Modular Forms, Vieweg 1994, p. 133.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Noam D. Elkies, Lattices, Linear Codes and Invariants, Part I, Notices of the Amer. Math. Soc., 47 (No. 10, Nov. 2000), 1238-1245, see p. 1242.
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A000122, A000700, A007331, A010054, A045823, A121373, .

Programs

Maple

q*product((1-q^(2*k-1))^8*(1-q^(4*k))^8, k=1..75);

Mathematica

a[0] = 0; a[n_] := -(-1)^n*Sum[ Mod[n/d, 2]*d^3, {d, Divisors[n]}]; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Jan 27 2012, after Michael Somos *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^8, {q, 0, n}]; (* Michael Somos, May 25 2014 *)

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A))^8, n))}; /* Michael Somos, Jul 16 2004 */

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (prod(k=1, n, (1 -( k%4==0) * x^k) * (1 - (k%2==1) * x^k), 1 + A))^8, n))}; /* Michael Somos, Jul 16 2004 */

PARI

{a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, (n/d%2) * d^3))}; /* Michael Somos, May 31 2005 */

Python

from sympy import divisors
def a(n): return 0 if n == 0 else -(-1)**n * sum([((n//d)%2) * d**3 for d in divisors(n)])
print([a(n) for n in range(101)])  # Indranil Ghosh, Jun 24 2017

Sage

A = ModularForms( Gamma0(4), 4, prec=70) . basis(); A[1] - 8*A[2] # _Michael Somos, May 25 2014

Formula

Expansion of (eta(q)* eta(q^4) / eta(q^2))^8 in powers of q. - Michael Somos, Jul 16 2004

Euler transform of period 4 sequence [-8, 0, -8, -8, ...]. - Michael Somos, Jul 16 2004

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = +u^4*w*v + 16*u^3*w*v^2 + 16*u^2*w^2*v^2 + 256*u^3*w^3 + 256*u^3*w^2*v + 4096*u^2*w^3*v + 4096*u*w^4*v + 4096*u*w^3*v^2 - u^2*v^4 - 16*u^2*w*v^3 - 256*u*w^2*v^3 - 256*w^2*v^4. - Michael Somos, May 31 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^4*u6^4 + u1^3*u2*u3^3*u6 + 2*u1*u2^3*u3*u6^3 - u2^4*u3^4.

Expansion of q * psi(-q)^8 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Mar 20 2008

a(n) is multiplicative with a(2^e) = -8^e if e>0, a(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1). - Michael Somos, Mar 20 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 16 (t/i)^4 f(t) where q = exp(2 Pi i t).

G.f.: x * (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(4*k)))^8.

a(n) = -(-1)^n * A007331(n).

a(2*n) = -8 * A007331(n). a(2*n + 1) = A045823(n). - Michael Somos, May 25 2014

Dirichlet g.f.: zeta(s-3) * zeta(s) * (1 - 1/2^s) * (1 - 1/2^(s-4)). - Amiram Eldar, Nov 03 2023