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Fungrim entry: d37d0f

μk=k1k+1(μk22+αk24)αk2μk1k+1   where αk={2,k=01,k=1j=2k1μjμk+1j,otherwise\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:kZ2k \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
LambertWPuiseuxCoefficientμk\mu_{k} Coefficient in scaled Puiseux expansion of Lambert W-function
Sumnf(n)\sum_{n} f(n) Sum
ZZGreaterEqualZn\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))))

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2020-04-08 16:14:44.404316 UTC