OFFSET
0,3
COMMENTS
This is the q-expansion corresponding to the vector [1, 0, 336, 768, 4950, 6912, 22944, 27648, 75792, 72192, 181728, 158976] in the space of modular forms on Gamma_1(12) with character Kronecker character -3 in modulus 12, weight 5, and dimension 11 over Integer Ring in the basis ordered by degree of leading term (as in Magma).
From Steven Kotlarz, Jun 11 2026: (Start)
The g.f. identity in the Formula section is proven: the theta series lies in M_5(Gamma_0(12), chi) with chi = Kronecker(-3, .) (Schoeneberg's theorem), each eta quotient lies in the same space (Ligozat's criteria), and the two sides agree to n=100, far beyond the Sturm bound of 40 for Gamma_1(12).
The eta-quotient terms of the formula pair up under the Atkin-Lehner involution W_3 with factors 3^4 and 3^6. (End)
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 157.
E. C. Pervin, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Andy Huchala, Table of n, a(n) for n = 0..20000
FORMULA
G.f.: (A1^8*A2^5/A6^3 + A1^2*A2^6*A3^2*A4*A6^2/A12^3 + 90*q*A1^4*A2*A3^4*A6 - 80*q*A1*A2*A3^5*A4^3*A6/A12 + 584*q^2*A1^2*A2*A3^2*A4^2*A6*A12^2 - 256*q^2*A2*A3^3*A4^5*A6*A12/A1 - 32*q^2*A1*A2^6*A4^2*A6^2*A12^2/A3^3 - 81*q^2*A1^2*A2^2*A3^2*A6^6*A12/A4^3 + 729*q^2*A3^8*A6^5/A2^3 - 128*q^3*A2*A4^4*A6*A12^4 + 2592*q^3*A2^2*A3*A4^2*A6^6*A12^2/A1^3)/2, where A_k = Product_{m>=1} (1 - q^(k*m)). - Steven Kotlarz, Jun 11 2026
EXAMPLE
G.f.: 1 + 336*q^2 + 768*q^3 + 4950*q^4 + 6912*q^5 + ...
The theta series itself is 1 + 336*x^4 + 768*x^6 + 4950*x^8 + ...; i.e., a(n) is the number of lattice vectors of squared norm 2*n
PROG
(SageMath)
e = DirichletGroup(12).1
M = ModularForms(e, 5, QQ)
bases = [_.q_expansion(20) for _ in M.integral_basis()]
list(sum(x*y for (x, y) in zip(bases, [1, 0, 336, 768, 4950, 6912, 22944, 27648, 75792, 72192, 181728, 158976]))) # Andy Huchala, Jun 05 2021
(PARI) my(q='q+O('q^102)); Vec((eta(q)^8*eta(q^2)^5/eta(q^6)^3 + eta(q)^2*eta(q^2)^6*eta(q^3)^2*eta(q^4)*eta(q^6)^2/eta(q^12)^3 + 90*q*eta(q)^4*eta(q^2)*eta(q^3)^4*eta(q^6) - 80*q*eta(q)*eta(q^2)*eta(q^3)^5*eta(q^4)^3*eta(q^6)/eta(q^12) + 584*q^2*eta(q)^2*eta(q^2)*eta(q^3)^2*eta(q^4)^2*eta(q^6)*eta(q^12)^2 - 256*q^2*eta(q^2)*eta(q^3)^3*eta(q^4)^5*eta(q^6)*eta(q^12)/eta(q) - 32*q^2*eta(q)*eta(q^2)^6*eta(q^4)^2*eta(q^6)^2*eta(q^12)^2/eta(q^3)^3 - 81*q^2*eta(q)^2*eta(q^2)^2*eta(q^3)^2*eta(q^6)^6*eta(q^12)/eta(q^4)^3 + 729*q^2*eta(q^3)^8*eta(q^6)^5/eta(q^2)^3 - 128*q^3*eta(q^2)*eta(q^4)^4*eta(q^6)*eta(q^12)^4 + 2592*q^3*eta(q^2)^2*eta(q^3)*eta(q^4)^2*eta(q^6)^6*eta(q^12)^2/eta(q)^3)/2) \\ Steven Kotlarz, Jun 11 2026
(Python)
def A006909_list(T):
def euler(d): # Product_{n>=1} (1-q^(d*n)), pentagonal number theorem
e = [0]*T; e[0] = 1; k = 1
while k*(3*k-1)//2*d < T:
s = -1 if k % 2 else 1
for g in (k*(3*k-1)//2*d, k*(3*k+1)//2*d):
if g < T: e[g] += s
k += 1
return e
def mul(a, b):
c = [0]*T
for i, ai in enumerate(a):
if ai:
for j in range(T-i):
if b[j]: c[i+j] += ai*b[j]
return c
def inv(a):
c = [0]*T; c[0] = 1
for n in range(1, T): c[n] = -sum(a[k]*c[n-k] for k in range(1, n+1))
return c
def pw(a, e):
if e < 0: a, e = inv(a), -e
r = [0]*T; r[0] = 1
while e:
if e & 1: r = mul(r, a)
a = mul(a, a); e >>= 1
return r
E = {d: euler(d) for d in (1, 2, 3, 4, 6, 12)}
terms = [(584, 2, {1:2, 2:1, 3:2, 4:2, 6:1, 12:2}), (90, 1, {1:4, 2:1, 3:4, 6:1}),
(-128, 3, {2:1, 4:4, 6:1, 12:4}), (-80, 1, {1:1, 2:1, 3:5, 4:3, 6:1, 12:-1}),
(-256, 2, {1:-1, 2:1, 3:3, 4:5, 6:1, 12:1}), (1, 0, {1:2, 2:6, 3:2, 4:1, 6:2, 12:-3}),
(-32, 2, {1:1, 2:6, 3:-3, 4:2, 6:2, 12:2}), (-81, 2, {1:2, 2:2, 3:2, 4:-3, 6:6, 12:1}),
(2592, 3, {1:-3, 2:2, 3:1, 4:2, 6:6, 12:2}), (1, 0, {1:8, 2:5, 6:-3}), (729, 2, {2:-3, 3:8, 6:5})]
total = [0]*T
for c, lead, rd in terms:
f = [0]*T; f[0] = 1
for d, r in rd.items(): f = mul(f, pw(E[d], r))
for i in range(T - lead): total[i + lead] += c*f[i]
return [t//2 for t in total]
print(A006909_list(26)) # Steven Kotlarz, Jun 11 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved