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Fungrim entry: d37d0f

μk=k1k+1(μk22+αk24)αk2μk1k+1   where α0=2,α1=1,αk=j=2k1μjμk+1j\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: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}_{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
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(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))), For(j, 2, Sub(k, 1)))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))))

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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