def _calc_bernoulli(n):
s = 0
a = int(C.binomial(n+3, n-6))
for j in xrange(1, n//6+1):
s += a * bernoulli(n - 6*j)
# Avoid computing each binomial coefficient from scratch
a *= _product(n-6 - 6*j + 1, n-6*j)
a //= _product(6*j+4, 6*j+9)
if n % 6 == 4:
s = -Rational(n+3, 6) - s
else:
s = Rational(n+3, 3) - s
return s / C.binomial(n+3, n)
def _calc_bernoulli(n):
s = 0
a = int(C.binomial(n + 3, n - 6))
for j in xrange(1, n // 6 + 1):
s += a * bernoulli(n - 6 * j)
# Avoid computing each binomial coefficient from scratch
a *= _product(n - 6 - 6 * j + 1, n - 6 * j)
a //= _product(6 * j + 4, 6 * j + 9)
if n % 6 == 4:
s = -Rational(n + 3, 6) - s
else:
s = Rational(n + 3, 3) - s
return s / C.binomial(n + 3, n)
def _eval_rewrite_as_Sum(self, arg):
if arg.is_even:
k = C.Dummy("k", integer=True)
j = C.Dummy("j", integer=True)
n = self.args[0] / 2
Em = (S.ImaginaryUnit * C.Sum( C.Sum( C.binomial(k,j) * ((-1)**j * (k-2*j)**(2*n+1)) /
(2**k*S.ImaginaryUnit**k * k), (j,0,k)), (k, 1, 2*n+1)))
return Em
def _eval_rewrite_as_Sum(self, arg):
if arg.is_even:
k = C.Dummy("k", integer=True)
j = C.Dummy("j", integer=True)
n = self.args[0] / 2
Em = (S.ImaginaryUnit * C.Sum( C.Sum( C.binomial(k, j) * ((-1)**j * (k - 2*j)**(2*n + 1)) /
(2**k*S.ImaginaryUnit**k * k), (j, 0, k)), (k, 1, 2*n + 1)))
return Em
def eval(cls, n, sym=None):
if n.is_Number:
if n.is_Integer and n.is_nonnegative:
if n is S.Zero:
return S.One
elif n is S.One:
if sym is None:
return -S.Half
else:
return sym - S.Half
# Bernoulli numbers
elif sym is None:
if n.is_odd:
return S.Zero
n = int(n)
# Use mpmath for enormous Bernoulli numbers
if n > 500:
p, q = bernfrac(n)
return Rational(int(p), int(q))
case = n % 6
highest_cached = cls._highest[case]
if n <= highest_cached:
return cls._cache[n]
# To avoid excessive recursion when, say, bernoulli(1000) is
# requested, calculate and cache the entire sequence ... B_988,
# B_994, B_1000 in increasing order
for i in xrange(highest_cached + 6, n + 6, 6):
b = cls._calc_bernoulli(i)
cls._cache[i] = b
cls._highest[case] = i
return b
# Bernoulli polynomials
else:
n, result = int(n), []
for k in xrange(n + 1):
result.append(C.binomial(n, k)*cls(k)*sym**(n - k))
return Add(*result)
else:
raise ValueError("Bernoulli numbers are defined only"
" for nonnegative integer indices.")
def _eval_rewrite_as_binomial(self,n):
return C.binomial(2*n,n)/(n + 1)
def _eval_rewrite_as_binomial(self, n):
return C.binomial(2 * n, n) / (n + 1)
