cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A354022 a(n) = n! * Sum_{d|n} mu(n/d) / d!.

Original entry on oeis.org

1, -1, -5, -11, -119, 241, -5039, -1679, -60479, 1784161, -39916799, 218877121, -6227020799, 43571848321, 1078831353601, -518918399, -355687428095999, 1058152455360001, -121645100408831999, 1115079416638387201, 42565648051390464001, 562000335730215782401
Offset: 1

Views

Author

Ilya Gutkovskiy, May 14 2022

Keywords

Links

Crossrefs

Cf. A008683, A057625, A062794, A068107, A099740, A132958, A327243, A332466.

Programs

  • Mathematica
    Table[n! Sum[MoebiusMu[n/d]/d!, {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[MoebiusMu[k] (Exp[x^k] - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n)=n! * sumdiv(n, d, moebius(n/d) / d!) \\ Winston de Greef, Sep 19 2023

Formula

E.g.f.: Sum_{k>=1} mu(k) * (exp(x^k) - 1).
Sum_{n>=1} a(n) * x^n / (n! * (1 - x^n)) = exp(x) - 1.

A286310 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - ...))))).

Original entry on oeis.org

1, 2, 14, 102, 944, 10378, 135134, 2026920, 34459410, 654728128, 13749310574, 316234132728, 7905853580624, 213458046541738, 6190283353628416, 191898783960483600, 6332659870762850624, 221643095476665302070, 8200794532637891559374, 319830986772877116086448
Offset: 1

Views

Author

Ilya Gutkovskiy, May 06 2017

Keywords

Examples

			G.f.: 1 + x/(1 - x) + 2*x^2/(1 - x^2) + 14*x^3/(1 - x^3) + 102*x^4/(1 - x^4) + ... = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - ...))))).

Links

Crossrefs

Cf. A001147, A002996, A062794.

Programs

  • Mathematica
    nn = 20; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-n x, 1, {n, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
a[n_] := Sum[MoebiusMu[n/d] (2 d - 1)!!, {d, Divisors[n]}]; Array[a, 20]

Formula

Sum_{d|n} a(d) = A001147(n) for n > 0.
a(n) ~ 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Sep 16 2021
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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