a.py

import sys

M = 10
cache = [[None for k in range(2**b+1)] for b in range(M)]

def maxo(a, b):
  if b is None:
    return a
  if a is None:
    return b
  return max(a, b)

def addo(a, b):
  if a is None or b is None:
    return None
  return a + b

def T(b,k):
  if k > 2 ** b:
    return None
  if cache[b][k] is not None:
    return cache[b][k]
  if b==0:
    return 0
  for l in range((k+1)//2, k+1):
    cache[b][k] = maxo(cache[b][k], addo(l, addo(T(b-1, l), T(b-1,k-l))))
  return cache[b][k]

for b in range(M):
  sys.stdout.write('b={}:'.format(b))
  for k in range(2 ** b + 1):
    sys.stdout.write(' {:3}'.format(T(b,k)))
  sys.stdout.write('\n')

for b in range(1,M):
  sys.stdout.write('T({:2},{:3})={:5}\n'.format(b, 2**(b-1), T(b,2**(b-1))))