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)
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)))