Fungrim entry: 588889

Table of

B_{n}\!\left(x\right)

for

0 \le n \le 10

$n$	$B_{n}\!\left(x\right)$
0	1
1	$x - \frac{1}{2}$
2	${x}^{2} - x + \frac{1}{6}$
3	${x}^{3} - \frac{3}{2} {x}^{2} + \frac{1}{2} x$
4	${x}^{4} - 2 {x}^{3} + {x}^{2} - \frac{1}{30}$
5	${x}^{5} - \frac{5}{2} {x}^{4} + \frac{5}{3} {x}^{3} - \frac{1}{6} x$
6	${x}^{6} - 3 {x}^{5} + \frac{5}{2} {x}^{4} - \frac{1}{2} {x}^{2} + \frac{1}{42}$
7	${x}^{7} - \frac{7}{2} {x}^{6} + \frac{7}{2} {x}^{5} - \frac{7}{6} {x}^{3} + \frac{1}{6} x$
8	${x}^{8} - 4 {x}^{7} + \frac{14}{3} {x}^{6} - \frac{7}{3} {x}^{4} + \frac{2}{3} {x}^{2} - \frac{1}{30}$
9	${x}^{9} - \frac{9}{2} {x}^{8} + 6 {x}^{7} - \frac{21}{5} {x}^{5} + 2 {x}^{3} - \frac{3}{10} x$
10	${x}^{10} - 5 {x}^{9} + \frac{15}{2} {x}^{8} - 7 {x}^{6} + 5 {x}^{4} - \frac{3}{2} {x}^{2} + \frac{5}{66}$

Table data:

\left(n, p\right)

such that

B_{n}\!\left(x\right) = p

Assumptions:

x \in \mathbb{C}

TeX:

x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("588889"),
    Description("Table of", BernoulliPolynomial(n, x), "for", LessEqual(0, n, 10)),
    Table(TableRelation(Tuple(n, p), Equal(BernoulliPolynomial(n, x), p)), TableHeadings(n, BernoulliPolynomial(n, x)), TableSplit(1), List(Tuple(0, 1), Tuple(1, Sub(x, Div(1, 2))), Tuple(2, Add(Sub(Pow(x, 2), x), Div(1, 6))), Tuple(3, Add(Sub(Pow(x, 3), Mul(Div(3, 2), Pow(x, 2))), Mul(Div(1, 2), x))), Tuple(4, Sub(Add(Sub(Pow(x, 4), Mul(2, Pow(x, 3))), Pow(x, 2)), Div(1, 30))), Tuple(5, Sub(Add(Sub(Pow(x, 5), Mul(Div(5, 2), Pow(x, 4))), Mul(Div(5, 3), Pow(x, 3))), Mul(Div(1, 6), x))), Tuple(6, Add(Sub(Add(Sub(Pow(x, 6), Mul(3, Pow(x, 5))), Mul(Div(5, 2), Pow(x, 4))), Mul(Div(1, 2), Pow(x, 2))), Div(1, 42))), Tuple(7, Add(Sub(Add(Sub(Pow(x, 7), Mul(Div(7, 2), Pow(x, 6))), Mul(Div(7, 2), Pow(x, 5))), Mul(Div(7, 6), Pow(x, 3))), Mul(Div(1, 6), x))), Tuple(8, Sub(Add(Sub(Add(Sub(Pow(x, 8), Mul(4, Pow(x, 7))), Mul(Div(14, 3), Pow(x, 6))), Mul(Div(7, 3), Pow(x, 4))), Mul(Div(2, 3), Pow(x, 2))), Div(1, 30))), Tuple(9, Sub(Add(Sub(Add(Sub(Pow(x, 9), Mul(Div(9, 2), Pow(x, 8))), Mul(6, Pow(x, 7))), Mul(Div(21, 5), Pow(x, 5))), Mul(2, Pow(x, 3))), Mul(Div(3, 10), x))), Tuple(10, Add(Sub(Add(Sub(Add(Sub(Pow(x, 10), Mul(5, Pow(x, 9))), Mul(Div(15, 2), Pow(x, 8))), Mul(7, Pow(x, 6))), Mul(5, Pow(x, 4))), Mul(Div(3, 2), Pow(x, 2))), Div(5, 66))))),
    Variables(x),
    Assumptions(Element(x, CC)))

Topics using this entry

Bernoulli numbers and polynomials