Revision history [back]

Hi

If I would like to generate my own symbolic math function : general_falling_factorial(x, j, n) . What I should add and modify in the code below ?

code on sageCell

j,n=var('j,n')
N=11


show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)= \
                \text{general_falling_factorial() is not the falling_factorial() SageMath Function} \
                \, = \, "),falling_factorial(x, n))


print "https://math.stackexchange.com/questions/497616/is-there-an-explicit-formula-for-the-bernoulli-numbers-that-doesnt-implicitly-r"
print "http://youngp.people.cofc.edu/factbern.pdf"

show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)"))




def general_falling_factorial_latex(self, x,j,n):
    return '{' + '('+str(i) + ' | ' + str(j) + ')'+'}_{' + str(n) + '}'
general_falling_factorial = function('general_falling_factorial', nargs=3, print_latex_func=general_falling_factorial_latex)



def general_falling_factorial(x, j, n):
    r""" to fill up """

    from sage.symbolic.expression import Expression
    from sage.structure.coerce import py_scalar_to_element
    x = py_scalar_to_element(x)
    j = py_scalar_to_element(j)
    n = py_scalar_to_element(n)

    if ((isinstance(j, Integer) or
        (isinstance(j, Expression) and
        j.is_integer())) and j >= 0) and \
        ((isinstance(n, Integer) or
        (isinstance(n, Expression) and
        n.is_integer())) and n >= 0)   :
        return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())


    return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())

def bernoulliExplicit(N) :
    bernoulliSL=[]
    for n in range(0,N) :
        bernoulliSLt=[]
        for j in range (1,n+2):
            #show("j : ",j)
            bernoulliSLt.append((((-1)^(j-1)/j) *(factorial(n+1))/((factorial(j)*(factorial((n+1-j))))))* \
                                sum([general_falling_factorial(i,j,n) for i in range(0,j)]))
        bernoulliSL.append(sum(bernoulliSLt))
    return bernoulliSL
show("Explicit Bernoulli List : ",bernoulliExplicit(N))


bernoulliL=[]
for i in range(0,N) :
    bernoulliL.append(bernoulli(i))


show("SageMath Bernoulli List : ",bernoulliL)

Hi

If I would like to generate my own symbolic math function : general_falling_factorial(x, j, n) . What I should add and modify in the code below ?? (Ido not know what to do with the two

   return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())
return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())

code on sageCell

j,n=var('j,n')
N=11


show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)= \
                \text{general_falling_factorial() is not the falling_factorial() SageMath Function} \
                \, = \, "),falling_factorial(x, n))


print "https://math.stackexchange.com/questions/497616/is-there-an-explicit-formula-for-the-bernoulli-numbers-that-doesnt-implicitly-r"
print "http://youngp.people.cofc.edu/factbern.pdf"

show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)"))




def general_falling_factorial_latex(self, x,j,n):
    return '{' + '('+str(i) + ' | ' + str(j) + ')'+'}_{' + str(n) + '}'
general_falling_factorial = function('general_falling_factorial', nargs=3, print_latex_func=general_falling_factorial_latex)



def general_falling_factorial(x, j, n):
    r""" to fill up """

    from sage.symbolic.expression import Expression
    from sage.structure.coerce import py_scalar_to_element
    x = py_scalar_to_element(x)
    j = py_scalar_to_element(j)
    n = py_scalar_to_element(n)

    if ((isinstance(j, Integer) or
        (isinstance(j, Expression) and
        j.is_integer())) and j >= 0) and \
        ((isinstance(n, Integer) or
        (isinstance(n, Expression) and
        n.is_integer())) and n >= 0)   :
        return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())


    return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())

def bernoulliExplicit(N) :
    bernoulliSL=[]
    for n in range(0,N) :
        bernoulliSLt=[]
        for j in range (1,n+2):
            #show("j : ",j)
            bernoulliSLt.append((((-1)^(j-1)/j) *(factorial(n+1))/((factorial(j)*(factorial((n+1-j))))))* \
                                sum([general_falling_factorial(i,j,n) for i in range(0,j)]))
        bernoulliSL.append(sum(bernoulliSLt))
    return bernoulliSL
show("Explicit Bernoulli List : ",bernoulliExplicit(N))


bernoulliL=[]
for i in range(0,N) :
    bernoulliL.append(bernoulli(i))


show("SageMath Bernoulli List : ",bernoulliL)

Hi

If I would like to generate my own symbolic math function : general_falling_factorial(x, j, n) . What I should add and modify in the code below ? (Ido not know what to do with the two

   return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())
return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())

code on sageCell

j,n=var('j,n')
N=11


show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)= \
                \text{general_falling_factorial() is not the falling_factorial() SageMath Function} \
                \, = \, "),falling_factorial(x, n))


print "https://math.stackexchange.com/questions/497616/is-there-an-explicit-formula-for-the-bernoulli-numbers-that-doesnt-implicitly-r"
print "http://youngp.people.cofc.edu/factbern.pdf"

show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)"))




def general_falling_factorial_latex(self, x,j,n):
    return '{' + '('+str(i) '('+str(x) + ' | ' + str(j) + ')'+'}_{' + str(n) + '}'
general_falling_factorial = function('general_falling_factorial', nargs=3, print_latex_func=general_falling_factorial_latex)



def general_falling_factorial(x, j, n):
    r""" to fill up """

    from sage.symbolic.expression import Expression
    from sage.structure.coerce import py_scalar_to_element
    x = py_scalar_to_element(x)
    j = py_scalar_to_element(j)
    n = py_scalar_to_element(n)

    if ((isinstance(j, Integer) or
        (isinstance(j, Expression) and
        j.is_integer())) and j >= 0) and \
        ((isinstance(n, Integer) or
        (isinstance(n, Expression) and
        n.is_integer())) and n >= 0)   :
        return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())


    return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())

def bernoulliExplicit(N) :
    bernoulliSL=[]
    for n in range(0,N) :
        bernoulliSLt=[]
        for j in range (1,n+2):
            #show("j : ",j)
            bernoulliSLt.append((((-1)^(j-1)/j) *(factorial(n+1))/((factorial(j)*(factorial((n+1-j))))))* \
                                sum([general_falling_factorial(i,j,n) for i in range(0,j)]))
        bernoulliSL.append(sum(bernoulliSLt))
    return bernoulliSL
show("Explicit Bernoulli List : ",bernoulliExplicit(N))


bernoulliL=[]
for i in range(0,N) :
    bernoulliL.append(bernoulli(i))


show("SageMath Bernoulli List : ",bernoulliL)

Hi

If I would like to generate my own symbolic math function : general_falling_factorial(x, j, n) . What I should add and modify in the code below ? (Ido not know what to do with the two

   return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())
return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())

code on sageCell

j,n=var('j,n')
N=11


show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)= \
                \text{general_falling_factorial() is not the falling_factorial() SageMath Function} \
                \, = \, "),falling_factorial(x, n))


print "https://math.stackexchange.com/questions/497616/is-there-an-explicit-formula-for-the-bernoulli-numbers-that-doesnt-implicitly-r"
print "http://youngp.people.cofc.edu/factbern.pdf"

show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)"))




def general_falling_factorial_latex(self, x,j,n):
    return '{' + '('+str(x) + ' | ' + str(j) + ')'+'}_{' + str(n) + '}'
general_falling_factorial = function('general_falling_factorial', nargs=3, print_latex_func=general_falling_factorial_latex)



def general_falling_factorial(x, j, n):
    r""" to fill up """

    from sage.symbolic.expression import Expression
    from sage.structure.coerce import py_scalar_to_element
    x = py_scalar_to_element(x)
    j = py_scalar_to_element(j)
    n = py_scalar_to_element(n)

    if ((isinstance(j, Integer) or
        (isinstance(j, Expression) and
        j.is_integer())) and j >= 0) and \
        ((isinstance(n, Integer) or
        (isinstance(n, Expression) and
        n.is_integer())) and n >= 0)   :
        return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())


    return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())

def bernoulliExplicit(N) :
    bernoulliSL=[]
    for n in range(0,N) :
        bernoulliSLt=[]
        for j in range (1,n+2):
            #show("j : ",j)
            bernoulliSLt.append((((-1)^(j-1)/j) *(factorial(n+1))/((factorial(j)*(factorial((n+1-j))))))* \
                                sum([general_falling_factorial(i,j,n) for i in range(0,j)]))
        bernoulliSL.append(sum(bernoulliSLt))
    return bernoulliSL
show("Explicit Bernoulli List : ",bernoulliExplicit(N))


bernoulliL=[]
for i in range(0,N) :
    bernoulliL.append(bernoulli(i))


show("SageMath Bernoulli List : ",bernoulliL)

Hi

If I would like to generate my own symbolic math function : general_falling_factorial(x, j, n) . What I should add and modify in the code below ? (Ido (I do not know what to do with the twotwo lines below I know it is ok for the first one, but what must I write for the second one ?)

   return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())
return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())

code on sageCell

j,n=var('j,n')
N=11


show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)= \
                \text{general_falling_factorial() is not the falling_factorial() SageMath Function} \
                \, = \, "),falling_factorial(x, n))


print "https://math.stackexchange.com/questions/497616/is-there-an-explicit-formula-for-the-bernoulli-numbers-that-doesnt-implicitly-r"
print "http://youngp.people.cofc.edu/factbern.pdf"

show(LatexExpr(r"B_n=\sum_{j=1}^{n+1}\frac{(-1)^{j-1}}{j}{n+1\choose j}\sum_{i=0}^{j-1}(i|j)_n"))
show(LatexExpr(r"\text{with }\, \,(i|j)_n:=i(i-j)\cdots(i-(n-1)j)"))




def general_falling_factorial_latex(self, x,j,n):
    return '{' + '('+str(x) + ' | ' + str(j) + ')'+'}_{' + str(n) + '}'
general_falling_factorial = function('general_falling_factorial', nargs=3, print_latex_func=general_falling_factorial_latex)



def general_falling_factorial(x, j, n):
    r""" to fill up """

    from sage.symbolic.expression import Expression
    from sage.structure.coerce import py_scalar_to_element
    x = py_scalar_to_element(x)
    j = py_scalar_to_element(j)
    n = py_scalar_to_element(n)

    if ((isinstance(j, Integer) or
        (isinstance(j, Expression) and
        j.is_integer())) and j >= 0) and \
        ((isinstance(n, Integer) or
        (isinstance(n, Expression) and
        n.is_integer())) and n >= 0)   :
        return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())


    return prod((((x)-(k*(j))) for k in (0..(n-1))), z=x.parent().one())

def bernoulliExplicit(N) :
    bernoulliSL=[]
    for n in range(0,N) :
        bernoulliSLt=[]
        for j in range (1,n+2):
            #show("j : ",j)
            bernoulliSLt.append((((-1)^(j-1)/j) *(factorial(n+1))/((factorial(j)*(factorial((n+1-j))))))* \
                                sum([general_falling_factorial(i,j,n) for i in range(0,j)]))
        bernoulliSL.append(sum(bernoulliSLt))
    return bernoulliSL
show("Explicit Bernoulli List : ",bernoulliExplicit(N))


bernoulliL=[]
for i in range(0,N) :
    bernoulliL.append(bernoulli(i))


show("SageMath Bernoulli List : ",bernoulliL)