Fungrim entry: 5ff181

{c}_{n} = \frac{1}{n {a}_{0}} \sum_{k=1}^{n} \left(\frac{k}{2} - n\right) {a}_{k} {c}_{n - k}\; \text{ where } {c}_{n} = [{x}^{n}] \frac{1}{\sqrt{A}},\;{a}_{n} = [{x}^{n}] A

Assumptions:

A \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] A \ne 0 \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

TeX:

{c}_{n} = \frac{1}{n {a}_{0}} \sum_{k=1}^{n} \left(\frac{k}{2} - n\right) {a}_{k} {c}_{n - k}\; \text{ where } {c}_{n} = [{x}^{n}] \frac{1}{\sqrt{A}},\;{a}_{n} = [{x}^{n}] A

A \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] A \ne 0 \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Sqrt`	$\sqrt{z}$	Principal square root
`PowerSeries`	$K[[x]]$	Formal power series
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("5ff181"),
    Formula(Where(Equal(Subscript(c, n), Mul(Div(1, Mul(n, Subscript(a, 0))), Sum(Mul(Mul(Sub(Div(k, 2), n), Subscript(a, k)), Subscript(c, Sub(n, k))), For(k, 1, n)))), Equal(Subscript(c, n), SeriesCoefficient(Div(1, Sqrt(A)), x, n)), Equal(Subscript(a, n), SeriesCoefficient(A, x, n)))),
    Variables(A, n),
    Assumptions(And(Element(A, PowerSeries(CC, x)), NotEqual(SeriesCoefficient(A, x, 0), 0), Element(n, ZZGreaterEqual(1)))))

Topics using this entry

Square roots