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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A386965 Number of permutations p of [2*n] such that there is at least one index i in [2*n-1] with p(i+1) = n + p(i).

Original entry on oeis.org

1, 10, 294, 16296, 1458120, 191751120, 34807535280, 8337722440320, 2547572372311680, 966944845408147200, 446304490431888211200, 246166572372916851532800, 159902551429370021259187200, 120818209587660157360960972800, 105060730670227917425027835648000
Offset: 1

Views

Author

Giovanni Resta, Aug 11 2025

Keywords

Comments

Problem 6 at IMO '89 essentially asks to show that a(n) > (2*n)!/4.

Examples

			The 10 permutations corresponding to a(2) are 1243, 1324, 1342, 2134, 2413, 2431, 3124, 3241, 4132, 4213.

Links

Crossrefs

Cf. A002674, A324361, A159960.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k+1) Binomial[n, k] (2 n - k)!, {k, n}]; Array[a, 15]

Formula

a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (2*n - k)!.
a(n) = (2*n)! * (1 - 1F1(-n; -2*n; -1)).
a(n) = n! * A324361(n).

A196080 Numerators of the sum of the n-th partial sums of the expansions of e and 1/e.

Original entry on oeis.org

2, 2, 3, 3, 37, 37, 1111, 1111, 6913, 6913, 799933, 799933, 739138093, 739138093, 44841044309, 44841044309, 32285551902481, 32285551902481, 9879378882159187, 9879378882159187, 1251387991740163687
Offset: 0

Views

Author

Paul Curtz, Sep 27 2011

Keywords

Comments

The n-th partial sums of the Taylor expansion of E are f(n) = A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, 163/60,.. .
The partial sums of an expansion of 1/e are essentially A000255(n-2)/A001048(n-1) preceded by 1 and 0, namely g(n)= 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760,... (Jolley's partial sums of 1/E in A068985 is the bisection 0, 1/3, 11/30, 103/280, 16687/45360,... of g(n).)
The current sequence are the numerators of f(n)+g(n), converging to E+1/E, namely 2, 2, 3, 3, 37/12, 37/12, 1111/360, 1111/360, 6913/2240 = 62217/21060, 6913/2240 = 62217/21060, 799933/259200 = 5599531/1814400,... The unreduced fractions are apparently given by duplicated A051396(n+1)/A002674(n).

Examples

			a(0)=1+1, a(1)=2+0, a(2)=(5+1)/2, a(3)=(8+1)/3.

Crossrefs

Cf. A001113, A068985, A137204 (e+1/e).

Programs

  • Mathematica
    a[n_] := (E*Gamma[n+1, 1] + (1/E)*Gamma[n+1, -1])/n! // FullSimplify // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 02 2012 *)

Extensions

Redefined by reduced fractions. - R. J. Mathar, Jul 02 2012

A256880 n*n!/round(n/2).

Original entry on oeis.org

1, 4, 9, 48, 200, 1440, 8820, 80640, 653184, 7257600, 73180800, 958003200, 11564467200, 174356582400, 2451889440000, 41845579776000, 671854030848000, 12804747411456000, 231125690776780800, 4865804016353280000
Offset: 1

Views

Author

M. F. Hasler, Apr 21 2015

Keywords

Crossrefs

Cf. A000142, A052145.

Programs

Formula

a(2n-1) = A052145(n).
a(2n) = 4*A002674(n) = 2*A010050(n) = 2^(n+1)*A000680(n), n>=1.
Previous Showing 31-33 of 33 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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