Fungrim entry: d37d0f

{\mu}_{k} = \frac{k - 1}{k + 1} \left(\frac{{\mu}_{k - 2}}{2} + \frac{{\alpha}_{k - 2}}{4}\right) - \frac{{\alpha}_{k}}{2} - \frac{{\mu}_{k - 1}}{k + 1}\; \text{ where } {\alpha}_{0} = 2,\,{\alpha}_{1} = -1,\,{\alpha}_{k} = \sum_{j=2}^{k - 1} {\mu}_{j} {\mu}_{k + 1 - j}

Assumptions:

k \in \mathbb{Z}_{\ge 2}

TeX:

{\mu}_{k} = \frac{k - 1}{k + 1} \left(\frac{{\mu}_{k - 2}}{2} + \frac{{\alpha}_{k - 2}}{4}\right) - \frac{{\alpha}_{k}}{2} - \frac{{\mu}_{k - 1}}{k + 1}\; \text{ where } {\alpha}_{0} = 2,\,{\alpha}_{1} = -1,\,{\alpha}_{k} = \sum_{j=2}^{k - 1} {\mu}_{j} {\mu}_{k + 1 - j}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`LambertWPuiseuxCoefficient`	${\mu}_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("d37d0f"),
    Formula(Where(Equal(LambertWPuiseuxCoefficient(k), Sub(Sub(Mul(Div(Sub(k, 1), Add(k, 1)), Add(Div(LambertWPuiseuxCoefficient(Sub(k, 2)), 2), Div(Subscript(alpha, Sub(k, 2)), 4))), Div(Subscript(alpha, k), 2)), Div(LambertWPuiseuxCoefficient(Sub(k, 1)), Add(k, 1)))), Equal(Subscript(alpha, 0), 2), Equal(Subscript(alpha, 1), -1), Equal(Subscript(alpha, k), Sum(Mul(LambertWPuiseuxCoefficient(j), LambertWPuiseuxCoefficient(Sub(Add(k, 1), j))), Tuple(j, 2, Sub(k, 1)))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))))

Topics using this entry

Lambert W-function