A396031 Growth series of higher Heisenberg group H_2, with standard generators.

1, 8, 40, 168, 546, 1512, 3642, 7800, 15108, 27320, 46620, 75720, 117590, 176664, 257790, 366536, 508664, 692472, 926544, 1220440, 1583514, 2029064, 2570674, 3222888, 3999404, 4919976, 6004180, 7272296, 8743694, 10445768, 12404582, 14647928, 17201648

Offset

0,2

Comments

Generating function is transcendental (Stoll, 1996).

Links

Table of n, a(n) for n=0..32.

Michael Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. math. 126, 85-109 (1996).

Prog

(Python)
from sympy import ImmutableMatrix, Matrix
def getGens(r):
# Construct the standard set of 2r generators and their inverses
   gens = []
   for i in range(r):
      gen = Matrix.eye(r+2)
      gen[i+1, r+1] = 1
      gens.append(ImmutableMatrix(gen))
      inv = Matrix.eye(r+2)
      inv[i+1, r+1] = -1
      gens.append(ImmutableMatrix(inv))
   for j in range(r):
      gen = Matrix.eye(r+2)
      gen[0, j+1] = 1
      gens.append(ImmutableMatrix(gen))
      inv = Matrix.eye(r+2)
      inv[0, j+1] = -1
      gens.append(ImmutableMatrix(inv))
   return gens
def growthHr(r, radius):
# Construct the growth sequence of H_r, up to the given radius
   ball = {ImmutableMatrix.eye(r+2)}
   sphere = {ImmutableMatrix.eye(r+2)}
   series = [1]
   gens = getGens(r)
   for _ in range(radius):
      new_sphere = set()
      for g in sphere:
         for gen in gens:
            newelt = g*gen
            if newelt not in ball:
               new_sphere.add(newelt)
      sphere = new_sphere
      ball.update(sphere)
      series.append(len(sphere))
   return series
print(growthHr(2, 13))

Crossrefs

Cf. A063810, A396032, A396033, A396034, A396035, A396036, A396037, A396038, A396039.

Sequence in context: A001789 A074412 A364619 * A217375 A113071 A006726

Adjacent sequences: A396028 A396029 A396030 * A396032 A396033 A396034

Keyword

nonn

Author

Alex Evetts, May 14 2026

Extensions

More terms from Sean A. Irvine, May 23 2026

Status

approved