def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return 2**(n + 1) * B / F * x**n
def taylor_term(n, x, *previous_terms):
from sympy import bernoulli
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return 2**(n + 1) * B/F * x**n
def taylor_term(n, x, *previous_terms):
"""
Returns the next term in the Taylor series expansion
"""
from sympy import bernoulli
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return 2 * (1 - 2**n) * B / F * x**n
def taylor_term(n, x, *previous_terms):
"""
Returns the next term in the Taylor series expansion
"""
from sympy import bernoulli
if n == 0:
return 1/sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return 2 * (1 - 2**n) * B/F * x**n
def gammaln_asymptotic_coefficient(k):
return bernoulli(2 * k + 2) / (2 * (k + 1) * (2 * k + 1))
def tetragamma_asymptotic_coefficient(k):
if k == 0:
return S(0)
return -(2 * k - 1) * bernoulli(2 * k - 2)
def trigamma_asymptotic_coefficient(k):
if k == 0:
return S(0)
return bernoulli(2 * k)
def test_bernoulli():
assert bernoulli(0) == 1
assert bernoulli(1) == Rational(-1, 2)
assert bernoulli(2) == Rational(1, 6)
assert bernoulli(3) == 0
assert bernoulli(4) == Rational(-1, 30)
assert bernoulli(5) == 0
assert bernoulli(6) == Rational(1, 42)
assert bernoulli(7) == 0
assert bernoulli(8) == Rational(-1, 30)
assert bernoulli(10) == Rational(5, 66)
assert bernoulli(1000001) == 0
assert bernoulli(0, x) == 1
assert bernoulli(1, x) == x - Rational(1, 2)
assert bernoulli(2, x) == x**2 - x + Rational(1, 6)
assert bernoulli(3, x) == x**3 - (3*x**2)/2 + x/2
# Should be fast; computed with mpmath
b = bernoulli(1000)
assert b.p % 10**10 == 7950421099
assert b.q == 342999030
b = bernoulli(10**6, evaluate=False).evalf()
assert str(b) == '-2.23799235765713e+4767529'
def test_bernoulli():
assert bernoulli(0) == 1
assert bernoulli(1) == Rational(-1, 2)
assert bernoulli(2) == Rational(1, 6)
assert bernoulli(3) == 0
assert bernoulli(4) == Rational(-1, 30)
assert bernoulli(5) == 0
assert bernoulli(6) == Rational(1, 42)
assert bernoulli(7) == 0
assert bernoulli(8) == Rational(-1, 30)
assert bernoulli(10) == Rational(5, 66)
assert bernoulli(1000001) == 0
assert bernoulli(0, x) == 1
assert bernoulli(1, x) == x - Rational(1, 2)
assert bernoulli(2, x) == x**2 - x + Rational(1, 6)
assert bernoulli(3, x) == x**3 - (3 * x**2) / 2 + x / 2
# Should be fast; computed with mpmath
b = bernoulli(1000)
assert b.p % 10**10 == 7950421099
assert b.q == 342999030
b = bernoulli(10**6, evaluate=False).evalf()
assert str(b) == '-2.23799235765713e+4767529'
def test_J1():
assert bernoulli(16) == R(-3617, 510)
def digamma_asymptotic_coefficient(k):
if k == 0:
return 0.0
return bernoulli(2 * k) / (2 * k)
def test_J1():
assert bernoulli(16) == R(-3617, 510)
t1 = edka.subs(sympy.besselk(2, k / t), sympy.Symbol('b2'))
t2 = t1.subs(sympy.besselk(3, k / t), sympy.Symbol('b3'))
print('edka = ' + str(sympy.simplify(t2).collect('b2')))
print('')
print('------------------------------------------------------')
print('Degenerate expansion')
print('')
z = sympy.Symbol('z', positive=True)
x = sympy.Symbol('x', positive=True)
pi = sympy.Symbol('pi', positive=True)
t = sympy.Symbol('t')
fz = (z * (2 + z))**(sympy.Rational(3, 2)) / 3
fz1 = sympy.diff(fz, z) * pi**2 * t**2 / 6
print(1, sympy.simplify(fz1.subs(z, x)))
print('')
fzt = fz1
for i in range(0, 6):
fzt = sympy.diff(fzt, z)
fzt = sympy.diff(fzt, z)
n = sympy.Rational(i + 2, 1)
term = (fzt.subs(z, x) * pi**(2 * n) * t**(2 * n) *
sympy.bernoulli(2 * n) * 2 * (2**(2 * n - 1) - 1) /
sympy.factorial(2 * n))
term = sympy.simplify(term)
print(i + 2, term)
print('')
#! /bin/python
from sympy import N, bernoulli, binomial
n = int(input())
for i in range(0, n + 1):
print(N(pow(-1, i) * bernoulli(i) * binomial(n + 1, i) / (n + 1)))
