class_docstr: Class for the Hypersphere Metric.
docstr: Compute the normalization factor - odd dimension.
docstr_tok: ["Compute","the","normalization","factor","-","odd","dimension","."]
code: def _normalization_factor_odd_dim(self, variances):
      """Compute the normalization factor - odd dimension."""
      dim = self._space.dim
      half_dim = int((dim + 1) / 2)
      area = 2 * gs.pi**half_dim / math.factorial(half_dim - 1)
      comb = gs.comb(dim - 1, half_dim - 1)
  
      erf_arg = gs.sqrt(variances / 2) * gs.pi
      first_term = (
          area
          / (2**dim - 1)
          * comb
          * gs.sqrt(gs.pi / (2 * variances))
          * gs.erf(erf_arg)
      )
  
      def summand(k):
          exp_arg = -((dim - 1 - 2 * k) ** 2) / 2 / variances
          erf_arg_2 = (
              gs.pi * variances - (dim - 1 - 2 * k) * 1j
          ) / gs.sqrt(2 * variances)
          sign = (-1.0) ** k
          comb_2 = gs.comb(k, dim - 1)
          return (
              sign
              * comb_2
              * gs.exp(exp_arg)
              * gs.real(gs.erf(erf_arg_2))
          )
  
      if half_dim > 2:
          sum_term = gs.sum(
              gs.stack([summand(k)] for k in range(half_dim - 2))
          )
      else:
          sum_term = summand(0)
      coef = (
          area / 2 / erf_arg * gs.pi**0.5 * (-1.0) ** (half_dim - 1)
      )
  
      return first_term + coef / 2 ** (dim - 2) * sum_term
  
imports: import logging
  import math
  from itertools import product
  from scipy.stats import beta
  import geomstats.algebra_utils
  import geomstats.backend
  from geomstats.geometry.base import LevelSet
  from geomstats.geometry.euclidean import Euclidean
  from geomstats.geometry.riemannian_metric import RiemannianMetric
  from geomstats.vectorization import get_batch_shape, repeat_out
label_desc: Use image node_id 11 for calling the HypersphereMetric obj's underlying member method code with example usage: obj._normalization_factor_odd_dim(variances) and returns: unknown, unknown