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_{k} = \begin{cases} 2, & k = 0\\-1, & k = 1\\\sum_{j=2}^{k - 1} \mu_{j} \mu_{k + 1 - j}, & \text{otherwise}\\ \end{cases}

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_{k} = \begin{cases} 2, & k = 0\\-1, & k = 1\\\sum_{j=2}^{k - 1} \mu_{j} \mu_{k + 1 - j}, & \text{otherwise}\\ \end{cases}

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

Definitions:

Fungrim symbol	Notation	Short description
`LambertWPuiseuxCoefficient`	$\mu_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function
`Sum`	$\sum_{n} f(n)$	Sum
`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(alpha_(Sub(k, 2)), 4))), Div(alpha_(k), 2)), Div(LambertWPuiseuxCoefficient(Sub(k, 1)), Add(k, 1)))), Def(alpha_(k), Cases(Tuple(2, Equal(k, 0)), Tuple(-1, Equal(k, 1)), Tuple(Sum(Mul(LambertWPuiseuxCoefficient(j), LambertWPuiseuxCoefficient(Sub(Add(k, 1), j))), For(j, 2, Sub(k, 1))), Otherwise))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))))

Topics using this entry

Lambert W-function