def monomial_count(V, N):
r"""
Computes the number of monomials.
The number of monomials is given by the following formula:
.. math::
\frac{(\#V + N)!}{\#V! N!}
where ``N`` is a total degree and ``V`` is a set of variables.
**Examples**
>>> from sympy import monomials, monomial_count
>>> from sympy.abc import x, y
>>> monomial_count(2, 2)
6
>>> M = monomials([x, y], 2)
>>> sorted(M)
[1, x, y, x**2, y**2, x*y]
>>> len(M)
6
"""
return C.Factorial(V + N) / C.Factorial(V) / C.Factorial(N)
def eval(cls, n, m, x):
if n.is_integer and n >= 0 and m.is_integer and abs(m) <= n:
assoc = cls.calc(int(n), abs(int(m)))
if m < 0:
assoc *= (-1)**(-m) * (C.Factorial(n + m)/C.Factorial(n - m))
return assoc.subs(_x, x)
if n.is_negative:
raise ValueError("%s : 1st index must be nonnegative integer (got %r)" % (cls, n))
if abs(m) > n:
raise ValueError("%s : abs('2nd index') must be <= '1st index' (got %r, %r)" % (cls, n, m))
def taylor_term(n, x, *previous_terms):
if n<0: return S.Zero
if n==0: return S.One
x = sympify(x)
if previous_terms:
p = previous_terms[-1]
if p is not None:
return p * x / n
return x**n/C.Factorial()(n)
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return -p * x**2 / (n * (n - 1))
else:
return (-1)**(n // 2) * x**(n) / C.Factorial(n)
def taylor_term(n, x, *previous_terms):
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = S.Bernoulli(n + 1)
F = C.Factorial(n + 1)
return (-1)**((n + 1) // 2) * 2**(n + 1) * B / F * x**n
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
a, b = ((n - 1) // 2), 2**(n + 1)
B = C.bernoulli(n + 1)
F = C.Factorial(n + 1)
return (-1)**a * b * (b - 1) * B / F * x**n
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
k = (n - 1) // 2
if len(previous_terms) > 2:
return -previous_terms[-2] * x**2 * (n - 2) / (n * k)
else:
return 2 * (-1)**k * x**n / (n * C.Factorial(k) * sqrt(S.Pi))
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
a = 2**(n + 1)
B = C.bernoulli(n + 1)
F = C.Factorial(n + 1)
return a * (a - 1) * B / F * x**n
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p * (n - 2)**2 / (n * (n - 1)) * x**2
else:
k = (n - 1) // 2
R = C.RisingFactorial(S.Half, k)
F = C.Factorial(k)
return R / F * x**n / n
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.Pi * S.ImaginaryUnit / 2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return p * (n - 2)**2 / (k * (k - 1)) * x**2
else:
k = (n - 1) // 2
R = C.RisingFactorial(S.Half, k)
F = C.Factorial(k)
return -R / F * S.ImaginaryUnit * x**n / n
