A098916 -id:A098916 - cp's OEIS frontend

A121723 a(n) = A098916(n+2) + (1-n) * A067318(n).

0, 3, 22, 150, 1096, 8820, 78408, 767088, 8212608, 95657760, 1205438400, 16350871680, 237633108480, 3685053415680, 60748282022400, 1061014235904000, 19574489449267200, 380408796994867200, 7768172642717491200

Offset: 1

Joel Duet, Aug 17 2006

nonn

This sequence arises when evaluating the generalized sub-volumes of the linearly weighted (n-1)-simplex in dimension n-1. For instance, in dimension 1 where n=2, the 1-simplex is the interval [H;J] of the real line (we suppose H < J). When H is weighted by the real h and J by j, the signed surface of the polygon {(H,0),(J,0),(J,j),(H,h)} of the Euclidean plane is S = (h+j)/2*(J-H).
Then we consider I the middle of [H;J]. It is linearly weighted by i = (h+j)/2. When we search for the weights w1(2) and w2(2) so that the 2 equations 2*Sh/(J-H) = h*w1(2) + j*w2(2) = (h+i)/2 and 2*Sj/(J-H) = h*w2(2) + j*w1(2) = (i+j)/2 are verified (which implies Sh + Sj = S also), we find that w1(2) = a(2)/A098916(2) = 3/4 and w2(2) = A067318(2)/A098916(2) = 1/4.
Even in higher dimensions (n > 2), there are only 2 weights: one for the considered sub-volume and the other for the other sub-volumes. For instance, in dimension 2 where n=3, the first weight w1(3) = 11/18 refers to the part of the triangle which is delimited by the 4 points: one top A, then the middle of [A;B], then the center of gravity, then the middle of [A;C]; and w2(3) = 7/36 refers to any of the 2 other parts of the triangle.

			a(3) = 22 because we can write 22 = A098916(3) + (1-3) * A067318(3) = 36 - 2*7.

Cf. A067318, A098916.

f[n_] := n! (n - 1) HarmonicNumber[n]; Array[f, 19] (* Robert G. Wilson v, Sep 07 2011 *)

a(n) = n!*(n-1)*Sum_{i=1..n} (1/i).

Definition corrected by Gary Detlefs, Sep 07 2011

A208529 Number of permutations of n > 1 having exactly 2 points on the boundary of their bounding square.

Original entry on oeis.org

2, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000

Offset: 2

David Nacin, Feb 27 2012

A bounding square for a permutation of n is the square with sides parallel to the coordinate axis containing (1,1) and (n,n), and the set of points P of a permutation p is the set {(k,p(k)) for 0 < k < n+1}.
Sequences A098558 and A052849 have the same terms except for the first. - Joerg Arndt, Mar 03 2012
a(n) is the number of permutations of n symbols that commute with a transposition: a permutation p of {1,...,n} has exactly two points on the boundary of their bounding square if and only if p commutes with transposition (1, n). - Luis Manuel Rivera Martínez, Feb 27 2014
a(n) is also the determinant of a matrix M each of whose elements M(i, j) is the result of a Reverse and Add operation (RADD) on i in base j: M(i,j) = i + (reverse(i) represented in base j), with 1 <= i < n and 1 < j <= n. - Federico Provvedi, May 10 2024

			a(2) = 2 because {(1,1),(2,2)} and {(1,2),(2,1)} each have two points on the bounding square.

Vincenzo Librandi, Table of n, a(n) for n = 2..200
Emeric Deutsch, Permutations and their bounding squares, Math Magazine, 85(1) (2012), 63.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Cf. A000142, A098916, A208528.

[2*Factorial(n)/n: n in [1..40]]; // Vincenzo Librandi, Apr 15 2014

A208529:=n->2*(n-2)!; seq(A208529(n), n=2..25); # Wesley Ivan Hurt, Feb 27 2014

Table[2(n-2)!, {n, 2, 10}]
FoldList[Times, 2, Range@21] (* Arkadiusz Wesolowski, May 08 2012 *)
Table[2 n!/n, {n, 1, 40}] (* Vincenzo Librandi, Apr 15 2014 *)

vector(33,n,2*n!/n) /* Anders Hellström, Jul 07 2015 */

import math
def a(n):
    return 2*math.factorial(n-2)

a(n) = 2*(n-2)!.
G.f.: G(0), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
G.f.: 2 + 2*x/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - x^2*(2*k+1)*(2*k+2)/( 1 - 2*x*(2*k+2) - x^2*(2*k+2)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 23 2013
a(n) = 2*n!/(n*(n-1)). - Vincenzo Librandi, Apr 15 2014
E.g.f.: 2 - (1 - x)*(2 + log(1/(1 - x)^2)). - Ilya Gutkovskiy, Mar 21 2018
Sum_{n>=2} 1/a(n) = e/2. - Amiram Eldar, Feb 02 2023

A208528 Number of permutations of n>1 having exactly 3 points P on the boundary of their bounding square.

Original entry on oeis.org

0, 4, 16, 72, 384, 2400, 17280, 141120, 1290240, 13063680, 145152000, 1756339200, 22992076800, 323805081600, 4881984307200, 78460462080000, 1339058552832000, 24186745110528000, 460970906812416000, 9245027631071232000, 194632160654131200000

Offset: 2

David Nacin, Feb 27 2012

nonn

A bounding square for a permutation of n is the square with sides parallel to the coordinate axis containing (1,1) and (n,n), and the set of points P of a permutation p is the set {(k,p(k)) for 0

a(n) is the number of permutations of n symbols that 3-commute with a transposition (see A233440 for definition): a permutation p of {1,...,n} has exactly three points on the boundary of their bounding square if and only if p 3-commutes with transposition (1, n). - Luis Manuel Rivera Martínez, Feb 27 2014

			a(3) = 4 because {(1,1),(2,3),(3,2)}, {(1,3),(2,1),(3,2)}, {(1,2),(2,3),(3,1)} and {(1,2),(2,1),(3,3)} each have three points on the bounding square.

Vincenzo Librandi, Table of n, a(n) for n = 2..200
Emeric Deutsch, Permutations and their bounding squares, Math Magazine, 85(1) (2012), p. 63.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Cf. A098916, A208529.

Cf. A001620, A091725, A099285.

Mathematica
```
Table[(4n-8)(n-2)!, {n, 2, 10}]
```

import math
def a(n):
    return (4*n-8)*math.factorial(n-2)

a(n) = (4*n-8) * (n-2)!
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=3} 1/a(n) = (Ei(1) - gamma)/4, where Ei(1) = A091725 and gamma = A001620.
Sum_{n>=3} (-1)^(n+1)/a(n) = (gamma - Ei(-1))/4, where Ei(-1) = -A099285. (End)

A153805 Decimal expansion of 3-e.

Original entry on oeis.org

2, 8, 1, 7, 1, 8, 1, 7, 1, 5, 4, 0, 9, 5, 4, 7, 6, 4, 6, 3, 9, 7, 1, 2, 5, 2, 8, 6, 4, 7, 3, 3, 7, 5, 0, 2, 2, 4, 2, 7, 5, 2, 9, 0, 6, 3, 0, 0, 0, 4, 0, 4, 2, 5, 0, 3, 3, 0, 3, 2, 3, 7, 2, 2, 7, 5, 9, 2, 3, 3, 6, 9, 6, 4, 6, 4, 5, 2, 4, 0, 5, 4, 2, 8, 6, 1, 7, 8, 2, 1, 4, 7, 4, 8, 3, 3, 5, 7, 2, 5, 7, 2, 5, 3

Offset: 0

Omar E. Pol, Jan 28 2009

			0.2817181715409547646397125286473375...

G. C. Greubel, Table of n, a(n) for n = 0..2000
Index entries for transcendental numbers

Cf. A001113, A052747, A098916. Essentially the same as A121239.

RealDigits[3 - E, 10, 150][[1]] (* Harvey P. Dale, Nov 29 2011 *)

From Amiram Eldar, Aug 06 2020: (Start)
Equals Sum_{k>=0} k/(k + 2)!.
Equals Sum_{k>=0} 1/(k! * (k + 2) * (k + 3)) = Sum_{k>=3} 1/A052747(k).
Equals Sum_{k>=2} 1/(k! * k * (k - 1)) = Sum_{k>=4} 1/A098916(k). (End)
Area between log(x) and log(x)^2, i.e., Integral_{x=1..exp(1)} (log(x) - log(x)^2) dx. - Martin Renner, May 17 2022

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A121723 a(n) = A098916(n+2) + (1-n) * A067318(n).

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A208529 Number of permutations of n > 1 having exactly 2 points on the boundary of their bounding square.

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A208528 Number of permutations of n>1 having exactly 3 points P on the boundary of their bounding square.

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A153805 Decimal expansion of 3-e.

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