def test_bell():
assert [bell(n) for n in range(8)] == [1, 1, 2, 5, 15, 52, 203, 877]
assert bell(0, x) == 1
assert bell(1, x) == x
assert bell(2, x) == x**2 + x
assert bell(5, x) == x**5 + 10*x**4 + 25*x**3 + 15*x**2 + x
X = symbols('x:6')
# X = (x0, x1, .. x5)
# at the same time: X[1] = x1, X[2] = x2 for standard readablity.
# but we must supply zero-based indexed object X[1:] = (x1, .. x5)
assert bell(6, 2, X[1:]) == 6*X[5]*X[1] + 15*X[4]*X[2] + 10*X[3]**2
assert bell(
6, 3, X[1:]) == 15*X[4]*X[1]**2 + 60*X[3]*X[2]*X[1] + 15*X[2]**3
X = (1, 10, 100, 1000, 10000)
assert bell(6, 2, X) == (6 + 15 + 10)*10000
X = (1, 2, 3, 3, 5)
assert bell(6, 2, X) == 6*5 + 15*3*2 + 10*3**2
X = (1, 2, 3, 5)
assert bell(6, 3, X) == 15*5 + 60*3*2 + 15*2**3
def test_bell():
assert [bell(n) for n in range(8)] == [1, 1, 2, 5, 15, 52, 203, 877]
assert bell(0, x) == 1
assert bell(1, x) == x
assert bell(2, x) == x**2 + x
assert bell(5, x) == x**5 + 10 * x**4 + 25 * x**3 + 15 * x**2 + x
X = symbols('x:6')
# X = (x0, x1, .. x5)
# at the same time: X[1] = x1, X[2] = x2 for standard readablity.
# but we must supply zero-based indexed object X[1:] = (x1, .. x5)
assert bell(6, 2,
X[1:]) == 6 * X[5] * X[1] + 15 * X[4] * X[2] + 10 * X[3]**2
assert bell(
6, 3,
X[1:]) == 15 * X[4] * X[1]**2 + 60 * X[3] * X[2] * X[1] + 15 * X[2]**3
X = (1, 10, 100, 1000, 10000)
assert bell(6, 2, X) == (6 + 15 + 10) * 10000
X = (1, 2, 3, 3, 5)
assert bell(6, 2, X) == 6 * 5 + 15 * 3 * 2 + 10 * 3**2
X = (1, 2, 3, 5)
assert bell(6, 3, X) == 15 * 5 + 60 * 3 * 2 + 15 * 2**3
def test_bell():
assert [bell(n) for n in range(8)] == [1, 1, 2, 5, 15, 52, 203, 877]
assert bell(0, x) == 1
assert bell(1, x) == x
assert bell(2, x) == x**2 + x
assert bell(5, x) == x**5 + 10 * x**4 + 25 * x**3 + 15 * x**2 + x
assert bell(oo) == S.Infinity
raises(ValueError, lambda: bell(oo, x))
raises(ValueError, lambda: bell(-1))
raises(ValueError, lambda: bell(S(1) / 2))
X = symbols('x:6')
# X = (x0, x1, .. x5)
# at the same time: X[1] = x1, X[2] = x2 for standard readablity.
# but we must supply zero-based indexed object X[1:] = (x1, .. x5)
assert bell(6, 2,
X[1:]) == 6 * X[5] * X[1] + 15 * X[4] * X[2] + 10 * X[3]**2
assert bell(
6, 3,
X[1:]) == 15 * X[4] * X[1]**2 + 60 * X[3] * X[2] * X[1] + 15 * X[2]**3
X = (1, 10, 100, 1000, 10000)
assert bell(6, 2, X) == (6 + 15 + 10) * 10000
X = (1, 2, 3, 3, 5)
assert bell(6, 2, X) == 6 * 5 + 15 * 3 * 2 + 10 * 3**2
X = (1, 2, 3, 5)
assert bell(6, 3, X) == 15 * 5 + 60 * 3 * 2 + 15 * 2**3
# Dobinski's formula
n = Symbol('n', integer=True, nonnegative=True)
# For large numbers, this is too slow
# For nonintegers, there are significant precision errors
for i in [0, 2, 3, 7, 13, 42, 55]:
assert bell(i).evalf() == bell(n).rewrite(Sum).evalf(subs={n: i})
# issue 9184
n = Dummy('n')
assert bell(n).limit(n, S.Infinity) == S.Infinity
def test_bell_sequence():
a = bell_sequence(20)
for n, b in enumerate(a):
assert b == bell(n)
def test_bell():
assert [bell(n) for n in range(8)] == [1, 1, 2, 5, 15, 52, 203, 877]
assert bell(0, x) == 1
assert bell(1, x) == x
assert bell(2, x) == x**2 + x
assert bell(5, x) == x**5 + 10*x**4 + 25*x**3 + 15*x**2 + x
X = symbols('x:6')
# X = (x0, x1, .. x5)
# at the same time: X[1] = x1, X[2] = x2 for standard readablity.
# but we must supply zero-based indexed object X[1:] = (x1, .. x5)
assert bell(6, 2, X[1:]) == 6*X[5]*X[1] + 15*X[4]*X[2] + 10*X[3]**2
assert bell(
6, 3, X[1:]) == 15*X[4]*X[1]**2 + 60*X[3]*X[2]*X[1] + 15*X[2]**3
X = (1, 10, 100, 1000, 10000)
assert bell(6, 2, X) == (6 + 15 + 10)*10000
X = (1, 2, 3, 3, 5)
assert bell(6, 2, X) == 6*5 + 15*3*2 + 10*3**2
X = (1, 2, 3, 5)
assert bell(6, 3, X) == 15*5 + 60*3*2 + 15*2**3
# Dobinski's formula
n = Symbol('n', integer=True, nonnegative=True)
# For large numbers, this is too slow
# For nonintegers, there are significant precision errors
for i in [0, 2, 3, 7, 13, 42, 55]:
assert bell(i).evalf() == bell(n).rewrite(Sum).evalf(subs={n: i})
# For negative numbers, the formula does not hold
m = Symbol('m', integer=True)
assert bell(-1).evalf() == bell(m).rewrite(Sum).evalf(subs={m: -1})
