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Fungrim entry: 5ff181

cn=1na0k=1n(k2n)akcnk   where cn=[xn]1A,an=[xn]A{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:AC[[x]]and[x0]A0andnZ1A \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
Sumnf(n)\sum_{n} f(n) Sum
Sqrtz\sqrt{z} Principal square root
FormalPowerSeriesK[[x]]K[[x]] Formal power series
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\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, FormalPowerSeries(CC, x)), Unequal(SeriesCoefficient(A, x, 0), 0), Element(n, ZZGreaterEqual(1)))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC