OFFSET
0,3
COMMENTS
The g.f. identity in the Formula section is proven: multiplying both sides by theta_3(q)^3, where theta_3(q) = A2^5/(A1^2*A4^2) is the theta series of sqrt(2)Z in this q^(norm/2) convention, turns both sides into modular forms of weight 6 on Gamma_0(16) with trivial character (the left side is the theta series of the even 12-dimensional lattice LAMBDA_9 + sqrt(2)Z^3 of determinant 4096, by Schoeneberg's theorem; each right-side term by Ligozat's criteria), and the two sides agree for all n <= 20000 (b-file) as well as for n <= 100 by direct lattice-vector enumeration from the Nebe-Sloane Gram matrix, far beyond the Sturm bound of 96 for Gamma_1(16). - Steven Kotlarz, Jun 13 2026
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 157.
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.: 9*A2^12*A4^10/(A1^8*A8^3*A16^2) - 6*A2^7*A4^10*A8^2/(A1^6*A16^4) - 28*q*A2^11*A8^10/(A1^6*A4^2*A16^4) + 40*q^2*A2^11*A8^4/A1^6 + 62*A2^2*A4^10*A8^7/(A1^4*A16^6) - 256*q^5*A2^10*A8^5*A16^6/(A1^4*A4^8) - 64*A4^10*A8^12/(A1^2*A2^3*A16^8) - 128*q*A4^12*A8^11/(A2^8*A16^6), where A_k = Product_{m>=1} (1 - q^(k*m)). - Steven Kotlarz, Jun 13 2026
EXAMPLE
G.f.: 1 + 272*q^2 + 256*q^3 + 3058*q^4 + 2048*q^5 + ...
The theta series itself is 1 + 272*x^4 + 256*x^6 + 3058*x^8 + ...; i.e., a(n) is the number of lattice vectors of squared norm 2*n.
PROG
(Magma)
L := Lattice("Lambda", 9);
B := Basis(ThetaSeriesModularFormSpace(L), 20);
Coefficients(B[1]+272*B[3]+256*B[4]+3058*B[5]+2048*B[6]+11232*B[7]+7168*B[8]+32848*B[9]+16384*B[10]); // Andy Huchala, Jun 16 2021
(PARI) T=102; q='q+O('q^T); E1=eta('q+O('q^T)); E2=eta('q^2+O('q^T)); E4=eta('q^4+O('q^T)); E8=eta('q^8+O('q^T)); E16=eta('q^16+O('q^T)); Vec((9*E2^12*E4^10/(E1^8*E8^3*E16^2) - 6*E2^7*E4^10*E8^2/(E1^6*E16^4) - 28*q*E2^11*E8^10/(E1^6*E4^2*E16^4) + 40*q^2*E2^11*E8^4/E1^6 + 62*E2^2*E4^10*E8^7/(E1^4*E16^6) - 256*q^5*E2^10*E8^5*E16^6/(E1^4*E4^8) - 64*E4^10*E8^12/(E1^2*E2^3*E16^8) - 128*q*E4^12*E8^11/(E2^8*E16^6))+O('q^T)) \\ Steven Kotlarz, Jun 13 2026
(Python)
def A005933_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, 4, 8, 16)}
terms = [(9, 0, {1:-8, 2:12, 4:10, 8:-3, 16:-2}), (-6, 0, {1:-6, 2:7, 4:10, 8:2, 16:-4}),
(-28, 1, {1:-6, 2:11, 4:-2, 8:10, 16:-4}), (40, 2, {1:-6, 2:11, 8:4}),
(62, 0, {1:-4, 2:2, 4:10, 8:7, 16:-6}), (-256, 5, {1:-4, 2:10, 4:-8, 8:5, 16:6}),
(-64, 0, {1:-2, 2:-3, 4:10, 8:12, 16:-8}), (-128, 1, {2:-8, 4:12, 8:11, 16:-6})]
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 total
print(A005933_list(26)) # Steven Kotlarz, Jun 13 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved