A396031 - OEIS

%I #22 Jun 09 2026 08:47:37

%S 1,8,40,168,546,1512,3642,7800,15108,27320,46620,75720,117590,176664,

%T 257790,366536,508664,692472,926544,1220440,1583514,2029064,2570674,

%U 3222888,3999404,4919976,6004180,7272296,8743694,10445768,12404582,14647928,17201648

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

%C Generating function is transcendental (Stoll, 1996).

%H Michael Stoll, <a href="https://doi.org/10.1007/s002220050090">Rational and transcendental growth series for the higher Heisenberg groups</a>, Invent. math. 126, 85-109 (1996).

%o (Python)

%o from sympy import ImmutableMatrix, Matrix

%o def getGens(r):

%o # Construct the standard set of 2r generators and their inverses

%o    gens = []

%o    for i in range(r):

%o       gen = Matrix.eye(r+2)

%o       gen[i+1,r+1] = 1

%o       gens.append(ImmutableMatrix(gen))

%o       inv = Matrix.eye(r+2)

%o       inv[i+1,r+1] = -1

%o       gens.append(ImmutableMatrix(inv))

%o    for j in range(r):

%o       gen = Matrix.eye(r+2)

%o       gen[0,j+1] = 1

%o       gens.append(ImmutableMatrix(gen))

%o       inv = Matrix.eye(r+2)

%o       inv[0,j+1] = -1

%o       gens.append(ImmutableMatrix(inv))

%o    return gens

%o def growthHr(r, radius):

%o # Construct the growth sequence of H_r, up to the given radius

%o    ball = {ImmutableMatrix.eye(r+2)}

%o    sphere = {ImmutableMatrix.eye(r+2)}

%o    series = [1]

%o    gens = getGens(r)

%o    for _ in range(radius):

%o       new_sphere = set()

%o       for g in sphere:

%o          for gen in gens:

%o             newelt = g*gen

%o             if newelt not in ball:

%o                new_sphere.add(newelt)

%o       sphere = new_sphere

%o       ball.update(sphere)

%o       series.append(len(sphere))

%o    return series

%o print(growthHr(2, 13))

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

%K nonn

%O 0,2

%A _Alex Evetts_, May 14 2026

%E More terms from _Sean A. Irvine_, May 23 2026