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Fungrim entry: 447541

[dndxnagm ⁣(1,x)]x=1=(1)nn!8nA060691 ⁣(n)\left[ \frac{d^{n}}{{d x}^{n}} \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = \frac{{\left(-1\right)}^{n} n !}{{8}^{n}} \text{A060691}\!\left(n\right)
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
  • Sequence A060691 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)
\left[ \frac{d^{n}}{{d x}^{n}} \operatorname{agm}\!\left(1, x\right) \right]_{x = 1} = \frac{{\left(-1\right)}^{n} n !}{{8}^{n}} \text{A060691}\!\left(n\right)

n \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
Powab{a}^{b} Power
Factorialn!n ! Factorial
SloaneAA00000X ⁣(n)\text{A00000X}\!\left(n\right) Sequence X in Sloane's OEIS
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(ComplexDerivative(AGM(1, x), For(x, 1, n)), Mul(Div(Mul(Pow(-1, n), Factorial(n)), Pow(8, n)), SloaneA("060691", n)))),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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

2021-03-15 19:12:00.328586 UTC