A052849 -id:A052849 - cp's OEIS frontend

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

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

A083410 a(n) = A083385(n)/n.

Original entry on oeis.org

1, 4, 22, 154, 1306, 12994, 148282, 1908274, 27333706, 431220034, 7428550042, 138737478994, 2792050329706, 60231133487074, 1386484468239802, 33921605427779314, 878976357571495306, 24046780495646314114, 692622345890928153562, 20950628198687114521234, 663992311200423614606506

Offset: 1

N. J. A. Sloane, Jun 08 2003

nonn

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of A052849(n+1)=[4,12,48,240,...] is 4*a(n)=[4,16,88,616,...].
Stirling transform of A001710(n+1)=[1,3,12,160,...] is a(n)=[1,4,22,154,...].
Stirling transform of A001563(n+1)=[4,18,96,600,...] is a(n+1)=[4,22,154,...]. (End)

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

A005649(n)=2*a(n), if n>0.
Pairwise sums of A091346.
Cf. A090665.

b:= proc(n, m) option remember;
     `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0)/2:
seq(a(n), n=1..23);  # Alois P. Heinz, Feb 14 2025

a[n_] := (-1)^n (PolyLog[-n - 1, 2] - PolyLog[-n, 2])/8;
Array[a, 21] (* Jean-François Alcover, Sep 10 2018, from A005649 *)

a(n)=if(n<0,0,n!*polcoeff(subst((1/(1-y)^2-1)/2,y,exp(x+x*O(x^n))-1),n))

E.g.f.: (1/(2-exp(x))^2-1)/2. - Michael Somos, Mar 04 2004
G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k+4) - 2*x^2*(k+1)*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n) = Sum_{k=1..n} k * A090665(n,k). - Alois P. Heinz, Feb 20 2025

A052578 a(0) = 0, a(n) = 4*n! for n > 0.

Original entry on oeis.org

0, 4, 8, 24, 96, 480, 2880, 20160, 161280, 1451520, 14515200, 159667200, 1916006400, 24908083200, 348713164800, 5230697472000, 83691159552000, 1422749712384000, 25609494822912000, 486580401635328000, 9731608032706560000, 204363768686837760000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..447
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 522

Cf. A000142, A052849 (k=2), A052560 (k=3).

[0] cat [4*Factorial(n): n in [1..25]]; // Vincenzo Librandi, Feb 13 2018

spec := [S,{S=Prod(Union(Z,Z,Z,Z),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0},4Range[25]!]  (* Harvey P. Dale, Mar 12 2011 *)

a(n)=if(n,4*n!,0) \\ Charles R Greathouse IV, Nov 20 2011

E.g.f.: -4*x/(-1+x)

Recurrence: {a(0)=0, (-1-n)*a(n)+a(n+1), a(1)=4}

A080046 Multiplicative Pascal triangle, read by rows: T(n,1)=T(n,n)=n and T(n,k) = T(n-1,k-1) * T(n-1,k).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 12, 12, 4, 5, 48, 144, 48, 5, 6, 240, 6912, 6912, 240, 6, 7, 1440, 1658880, 47775744, 1658880, 1440, 7, 8, 10080, 2388787200, 79254226206720, 79254226206720, 2388787200, 10080, 8, 9, 80640, 24078974976000

Offset: 1

Reinhard Zumkeller, Jan 21 2003

T(n,2) = A052849(n) for n>1.

Reinhard Zumkeller and Andre F. Labossiere, Rows n=1..14 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A082611.

Cf. A051597.

a080046 n k = a080046_tabl !! (n-1) !! (k-1)
a080046_row n = a080046_tabl !! (n-1)
a080046_tabl = iterate f [1] where
   f (x:xs) = [x + 1] ++ (zipWith (*) xs $ reverse xs) ++ [x + 1]
-- Reinhard Zumkeller, Oct 27 2013

Corrected by André F. Labossière, Sep 27 2004

A285201 Stage at which Ken Knowlton's elevator (version 1) reaches floor n for the first time.

Original entry on oeis.org

1, 2, 5, 14, 45, 174, 825, 4738, 32137, 251338, 2224157, 21952358, 238962581, 2843085270, 36696680241, 510647009850, 7619901954001, 121367981060434, 2055085325869813, 36861997532438654, 698193329457246653, 13924819967953406654, 291683979376372766697, 6402385486361598687666, 146948520147021794869977

Offset: 1

R. L. Graham, May 02 2017

nonn

Indices of records in A285200.

When prefixed by a(0)=0, the first differences give A111063. - N. J. A. Sloane, May 03 2017

N. J. A. Sloane, Table of n, a(n) for n = 1..400
Ken Knowlton, Illustration showing what floor the elevator is on for the first 49 stages

Cf. A285200, A285203, A111063, A098558, A052849, A286281, A286282.

a:= proc(n) option remember; `if`(n<3, n, ((n-1)^2*a(n-1)
      -(n-2)*(2*n-3)*a(n-2)+(n-1)*(n-3)*a(n-3))/(n-2))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 11 2018

a[n_] := 2 - n + 2 Sum[k!/j!, {k, 0, n-2}, {j, 0, k}];
Array[a, 25] (* Jean-François Alcover, Nov 01 2020 *)

a(n) = 2 - n + 2 * Sum_{k=0..n-2} Sum_{j=0..k} k!/j!.
For n >= 2, a(n) = 1+n+2*Sum_{k=2..n} C(n,k)*(k-1)! = 1+n+2*n!*Sum_{k=2..n} 1/(k*(n-k)!). - N. J. A. Sloane, May 03 2017
E.g.f.: exp(x)*(1-x-2*log(1-x)). Omitting the factor exp(x), this gives (essentially) the e.g.f. for A098558 (or A052849). - N. J. A. Sloane, May 03 2017

A110141 Triangle, read by rows, where row n lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k).

Original entry on oeis.org

1, 1, 2, 2, 6, 2, 3, 24, 4, 3, 8, 4, 120, 12, 6, 8, 4, 6, 5, 720, 48, 18, 16, 8, 6, 5, 48, 8, 18, 6, 5040, 240, 72, 48, 24, 12, 10, 48, 8, 18, 6, 24, 10, 12, 7, 40320, 1440, 360, 192, 96, 36, 30, 96, 16, 36, 12, 24, 10, 12, 7, 384, 32, 36, 12, 15, 32, 8, 362880, 10080, 2160, 960

Offset: 0

Paul D. Hanna, Jul 13 2005

Row n starts with n!, after which the following pattern holds. When terms of row n are divided by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1, the result is the initial terms of A110142. E.g., row 6 is: {720,48,18,16,8,6,5,48,8,18,6}; divide by respective factorials: {6!,4!,3!,2!,2!,1!,1!,0!,0!,0!,0!} with {4!,3!,2!,1!,0!} respectively occurring {1,1,2,2,4} times (A002865), yields the initial terms of A110142: {1,2,3,8,4,6,5,48,8,18,6}.
The term of the sequence corresponding to the product c_1^{n_1}c_2^{n_2}...c_k^{n_k} is equal to the number of elements in the centralizer of a permutation of n_1+2n_2+...+kn_k elements whose cycle type is 1^{n_1}2^{n_2}...k^{n^k}. (This fact is very standard, in particular, for the theory of symmetric functions.) - Vladimir Dotsenko, Apr 19 2009
Multiplying the values of row n by the corresponding values in row n of A102189, one obtains n!. - Jaimal Ichharam, Aug 06 2015
a(n,k) is the number of permutations in S_n that commute with a permutation having cycle type "k".  Here, the cycle type of an n-permutation pi is the vector (i_1,...,i_n) where i_j is the number of cycles in pi of length j.  These A000041(n) vectors can be ordered in reverse lexicographic order.  The k-th cycle type is the k-th vector in this ordering. - Geoffrey Critzer, Jan 18 2019

			Coefficients [x^n] exp(c1*x + (c2/2)*x^2 + (c3/3)*x^3 + ...) begin:
[x^0]: 1;
[x^1]: 1*c1;
[x^2]: (1/2)*c1^2 + (1/2)*c2;
[x^3]: (1/6)*c1^3 + (1/2)*c1*c2 + (1/3)*c3;
[x^4]: (1/24)*c1^4 + (1/4)*c1^2*c2 + (1/3)*c1*c3 + (1/8)*c2^2 + (1/4)*c4;
[x^5]: (1/120)*c1^5 + (1/12)*c1^3*c2 + (1/6)*c1^2*c3 + (1/8)*c1*c2^2 + (1/4)*c1*c4 + (1/6)*c2*c3 + (1/5)*c5;
[x^6]: (1/720)*c1^6 + (1/48)*c1^4*c2 + (1/18)*c1^3*c3 + (1/16)*c1^2*c2^2 + (1/8)*c1^2*c4 + (1/6)*c1*c2*c3 + (1/5)*c1*c5 + (1/48)*c2^3 + (1/8)*c2*c4 + (1/18)*c3^2 + (1/6)*c6;
forming this triangle of unit fraction coefficients:
1;
1;
2,2;
6,2,3;
24,4,3,8,4;
120,12,6,8,4,6,5;
720,48,18,16,8,6,5,48,8,18,6;
5040,240,72,48,24,12,10,48,8,18,6,24,10,12,7;
40320,1440,360,192,96,36,30,96,16,36,12,24,10,12,7,384,32,36,12,15,32,8;
362880,10080,2160,960,480,144,120,288,48,108,36,48,20,24,14,384,32,36,12,15,32,8,144,40,24,14,162,18,20,9; ...

Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford University Press, 1995. [From Vladimir Dotsenko, Apr 19 2009]

Cf. A000041, A002865, A102189, A110142, A110143 (row sums).

First, second and third entries of each row are given (up to an offset) by A000142, A052849, and A052560 respectively. - Vladimir Dotsenko, Apr 19 2009

Table[n!/CoefficientRules[n! CycleIndex[SymmetricGroup[n], s]][[All, 2]], {n, 1, 8}] // Grid (* Geoffrey Critzer, Jan 18 2019 *)

Number of terms in row n is A000041(n) (partition numbers). The unit fractions of each row sum to unity: Sum_{k=1..A000041(n)} 1/T(n, k) = 1.

a(n,k) = n!/A181897(n,k). - Geoffrey Critzer, Jan 18 2019

A275845 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(a(n)).

Original entry on oeis.org

0, 1, 4, 2, 6, 3, 12, 8, 16, 5, 15, 10, 18, 21, 22, 7, 20, 13, 24, 27, 28, 9, 26, 17, 48, 30, 52, 33, 34, 11, 60, 32, 64, 23, 56, 36, 66, 61, 70, 39, 40, 14, 73, 38, 78, 29, 67, 42, 72, 80, 76, 74, 85, 45, 84, 46, 88, 19, 89, 44, 90, 97, 94, 35, 81, 49, 87, 99, 93, 91, 105, 53, 96, 104, 100, 54, 109, 25, 108, 110, 112, 51, 111, 121, 114, 117, 118, 41

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275846.
Cf. A000142, A052849, A153880, A225901, A255411, A256450, A257680, A266193, A273663, A273670.
Similar permutations: A273667 (a more recursed variant), A275847, A275848.

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [when n is one of the terms of A153880] then a(n) = A255411(A266193(n)), otherwise [when n is one of the terms of A273670], a(n) = A256450(a(A273663(n))).
Other identities:
a(A000142(n)) = A052849(n) for all n >= 2.

A351893 Numbers that contain only even digits in their factorial-base representation.

Original entry on oeis.org

0, 4, 12, 16, 48, 52, 60, 64, 96, 100, 108, 112, 240, 244, 252, 256, 288, 292, 300, 304, 336, 340, 348, 352, 480, 484, 492, 496, 528, 532, 540, 544, 576, 580, 588, 592, 1440, 1444, 1452, 1456, 1488, 1492, 1500, 1504, 1536, 1540, 1548, 1552, 1680, 1684, 1692, 1696

Offset: 1

Amiram Eldar, Feb 24 2022

All the terms are multiples of 4 (A008586).

			4 is a term since its factorial-base presentation, 20, has only even digits.
16 is a term since its factorial-base presentation, 220, has only even digits.

Amiram Eldar, Table of n, a(n) for n = 1..14400 (terms below 10!)
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Cf. A007623, A008586, A351894.
Subsequence: A052849 \ {2}.
Similar sequences: A005823 (ternary), A014263 (decimal), A062880 (quaternary).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[0, max!, 2], AllTrue[fctBaseDigits[#], EvenQ] &]

A052648 Expansion of e.g.f. 5*x/(1-x).

Original entry on oeis.org

0, 5, 10, 30, 120, 600, 3600, 25200, 201600, 1814400, 18144000, 199584000, 2395008000, 31135104000, 435891456000, 6538371840000, 104613949440000, 1778437140480000, 32011868528640000, 608225502044160000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 595
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Cf. A000142, A052849 (k=2), A052560 (k=3), A052578 (k=4).

spec := [S,{S=Prod(Sequence(Z),Union(Z,Z,Z,Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(5x)/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 01 2016 *)

E.g.f.: 5*x/(1-x)
Recurrence: {a(0)=0, (-1-n)*a(n)+a(n+1)=0, a(1)=5}
a(n) = 5*n!, n>0.

A086915 Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).

Original entry on oeis.org

2, 4, 4, 12, 24, 8, 48, 144, 96, 16, 240, 960, 960, 320, 32, 1440, 7200, 9600, 4800, 960, 64, 10080, 60480, 100800, 67200, 20160, 2688, 128, 80640, 564480, 1128960, 940800, 376320, 75264, 7168, 256, 725760, 5806080, 13547520, 13547520, 6773760, 1806336

Offset: 1

Vladeta Jovovic, Sep 24 2003

Also the Bell transform of A052849(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

The coefficients of n! * L_n(-2*x,-1), where n! * L_n(-x,-1) are the normalized, unsigned Laguerre polynomials of order -1 of A105278, also known as the Lah polynomials, which are also a shifted version of n! * L_n(-x,1). Cf. p. 8 of the Gross and Matytsin link. - Tom Copeland, Sep 30 2016

			Triangle begins:
   2;
   4,   4;
  12,  24,  8;
  48, 144, 96, 16;
  ...

G. C. Greubel, Rows n=1..100 of the triangle, flattened
D. Gross and A. Matytsin, Instanton induced large N phase transitions in two and four dimensional QCD, arXiv:hep-th/9404004, 1994.
Index entries for sequences related to Laguerre polynomials

Cf. A008297, A052897 (row sums), A059110, A079621, A105278.

[Factorial(n)*Binomial(n-1,k-1)*2^k/Factorial(k): k in [1..n], n in [1..10]]; // G. C. Greubel, May 23 2018

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> 2*(n+1)!, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[n!/k! Binomial[n-1,k-1]2^k,{n,10},{k,n}]] (* Harvey P. Dale, May 25 2011 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2*(#+1)!&, rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

for(n=1,10, for(k=1, n, print1(n!/k!*binomial(n-1,k-1)*2^k, ", "))) \\ G. C. Greubel, May 23 2018

E.g.f.: exp(2*x*y/(1-x)).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (-2)^k * A008297(n, k) = 2^k * A105278(n, k).
Sum_{k=1..n} T(n, k) = 2 * n! * Hypergeometric1F1([1-n], [2], -2) = 2*(n-1)! * LaguerreL(n-1, 1, -2) = A253286(n, 2). (End)

cp's OEIS Frontend

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

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A083410 a(n) = A083385(n)/n.

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A052578 a(0) = 0, a(n) = 4*n! for n > 0.

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A080046 Multiplicative Pascal triangle, read by rows: T(n,1)=T(n,n)=n and T(n,k) = T(n-1,k-1) * T(n-1,k).

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A285201 Stage at which Ken Knowlton's elevator (version 1) reaches floor n for the first time.

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A110141 Triangle, read by rows, where row n lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k).

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A275845 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(a(n)).

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