def bell(n):
if n < 0:
raise ValueError('Bell polynomials only defined for n >= 0')
if n == 0:
return ONE
if n == 1:
return x
return sum(stirling2(n, k) * x ** k for k in xrange(n + 1))
def testStirling2(self):
py.test.raises(ValueError, lambda: funcs.stirling2(1, -1))
assert funcs.stirling2(0, 0) == 1
assert funcs.stirling2(2, 1) == 1
assert funcs.stirling2(12, 3) == 86526
assert funcs.stirling2(13, 13) == 1
assert funcs.stirling2(13, 12) == 78
def touchard(n):
'''
Returns the *n-th* Touchard polynomial in ``x``.
:rtype: :class:`pypol.Polynomial`
**Examples**
::
>>> touchard(0)
+ 1
>>> touchard(1)
+ x
>>> touchard(2)
+ x^2 + x
>>> touchard(12)
+ x^12 + 66x^11 + 7498669301432319/4398046511104x^10 + 22275x^9 + 159027x^8 + 627396x^7 + 1323652x^6 + 1379400x^5 + 611501x^4 + 86526x^3 + 2047x^2 + x
The Touchard polynomials also satisfy:
:math:`T_n(1) = B_n`
where :math:`B_n` is the *n-th* Bell number (:func:`funcs.bell_num`)::
>>> long(touchard(19)(1)) == long(bell_num(19))
True
The more *n* become greater, the more it loses precision::
>>> long(touchard(23)(1)) == long(bell_num(23))
False
>>> abs(long(touchard(23)(1)) - long(bell_num(23)))
8L
>>> long(touchard(45)(1)) == long(bell_num(45))
False
>>> abs(long(touchard(45)(1)) - long(bell_num(45)))
19342813113834066795298816L
>>> long(touchard(123)(1)) == long(bell_num(123))
False
>>> abs(long(touchard(123)(1)) - long(bell_num(123)))
429106803807439187983719223678319701219747465049443431177466446916319867062128867811451292833717675081551803755143128885524389827706879L
'''
if n < 0:
return NULL
if n == 0:
return ONE
return sum(stirling2(n, k) * x ** k for k in xrange(n + 1))
