A233440 -id:A233440 - cp's OEIS frontend

A004320 a(n) = n(n+1)(n+2)^2/6.

0, 3, 16, 50, 120, 245, 448, 756, 1200, 1815, 2640, 3718, 5096, 6825, 8960, 11560, 14688, 18411, 22800, 27930, 33880, 40733, 48576, 57500, 67600, 78975, 91728, 105966, 121800, 139345, 158720, 180048, 203456, 229075, 257040, 287490, 320568, 356421, 395200

Offset: 0

N. J. A. Sloane

Consider the set B(n) = {1,2,3,...n}. Let a(0) = 0. Then a(n) = Sum [ b(i)^2 - b(j)^2] for all i, j = 1 to n, b(i) belongs to B(n). E.g., a(3) = (3^2-1^2) + (3^2-2^2) + (2^2-1^2) = 16. - Amarnath Murthy, Jun 01 2001
Partial sums of A016061. - J. M. Bergot, Jun 18 2013
For n >= 3, a(n-2) is the number of permutations of n symbols that 3-commute with an n-cycle (see A233440 for definition). - Luis Manuel Rivera Martínez, Feb 24 2014
a(n) is the sum of all pairs with repetitions allowed drawn from the set of triangular numbers from A000217(0) to A000217(n). This is similar to A027480 but uses triangular numbers instead of the integers. Example for n=2: 0+1, 0+3, 1+1, 1+3, 3+3 gives sum of 16 = a(2). - J. M. Bergot, Mar 23 2016
From Mircea Dan Rus, Jul 29 2020: (Start)
a(n) is the number of lattice rectangles (squares included) inside half of an Aztec diamond of order n. This shape is obtained by stacking n rows of consecutive unit lattice squares, with the centers of rows vertically aligned and consisting successively of 2n, 2n-2,..., 4, 2 squares. See below the representation for n=6.
              
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(End)
a(n-1) = (n+1)*binomial(n+1, 3) is the number of certain rectangles (squares included) in an n X n square filled with 1 X 1 squares. Divide the n X n square, for n >= 2, into two complementary staircases by the boundary consisting of 2*n length 1 edges. For n = 1 there is no boundary. See a A000332 figure in the Mircea Dan Rus comment for the staircase with basis length n = 4. The complementary staircase is upside down with basis length n-1 = 3. Then a(n-1) is the number of rectangles in the n X n square which have at least one border link in their interior. This counting is based on the binomial identity given in the formula section, using A096948 (for n=m), A000332(n+3) and A000332(n+2). - Wolfdieter Lang, Sep 22 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Teofil Bogdan and Mircea Dan Rus, Counting the lattice rectangles inside Aztec diamonds and square biscuits, arXiv:2007.13472 [math.CO], 2020.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A016061, A047929, A233440, A000332, A096948, A000217.

[n*(n+1)*(n+2)^2/6: n in [0..40] ]; // Vincenzo Librandi, Aug 19 2011

[seq ((n+2)*(binomial(n+2,3)), n=0..45)]; # Zerinvary Lajos, May 26 2006

Table[n (n + 1) (n + 2)^2/6, {n, 0, 40}] (* Wesley Ivan Hurt, Oct 28 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{0,3,16,50,120,245},40] (* Harvey P. Dale, Nov 09 2022 *)

a(n)=n*(n+1)*(n+2)^2/6 \\ Charles R Greathouse IV, Jun 18 2013

G.f.: x*(3+x)/(1-x)^5. - Paul Barry, Feb 27 2003
a(n) = (n+2)*A000292(n). - Zerinvary Lajos, May 26 2006
a(n) = A047929(n+2)/6. - Zerinvary Lajos, May 09 2007
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Oct 28 2014
a(n) = 3*A000332(n+3) + A000332(n+2). - Mircea Dan Rus, Jul 29 2020
Sum_{n>=1} 1/a(n) = Pi^2/2 - 9/2. - Jaume Oliver Lafont, Jul 13 2017
a(n-1) = T(n)^2 - (s(n) + s(n-1)), with T(n) = binomial(n+1, 2) = A000217(n) and s(n) = binomial(n+3, 4) = A000332(n+3), for n >= 1. See a comment above, and the formula by Mircea Dan Rus. - Wolfdieter Lang, Sep 22 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/4 + 12*log(2) - 21/2. - Amiram Eldar, Jan 28 2022
E.g.f.: exp(x)*x*(18 + 30*x + 11*x^2 + x^3)/6. - Stefano Spezia, Mar 04 2023
a(n) = Sum_{j=0..n+1} binomial(n+1,2) + binomial(n+1,3). - Detlef Meya, Jan 20 2024

A047929 a(n) = n^2(n-1)(n-2).

Original entry on oeis.org

0, 18, 96, 300, 720, 1470, 2688, 4536, 7200, 10890, 15840, 22308, 30576, 40950, 53760, 69360, 88128, 110466, 136800, 167580, 203280, 244398, 291456, 345000, 405600, 473850, 550368, 635796, 730800, 836070, 952320, 1080288, 1220736

Offset: 2

N. J. A. Sloane

There are 5 ways to put parentheses in the expression a - b - c - d: (a - (b - c)) - d, ((a - b) - c) - d, (a - b) - (c - d), a - (b - (c - d)), a - ((b - c) - d). This sequence describes how many sets of natural numbers [a,b,c,d] can be produced with the numbers {0,1,2,3,...,n} such that all the distinct expressions take different values. A045991 describes the similar process for a - b - c. For example, no sets can be produced with only 0's or only 0's and 1's; with {0,1,2,3}, 18 such sets can be produced. - Asher Auel, Jan 26 2000

For n >= 3, a(n)/6 is the number of permutations of n symbols that 3-commute with an n-cycle (see A233440 for definition). - Luis Manuel Rivera Martínez, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for sequences related to parenthesizing
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A004320, A045991, A233440.

[n^2*(n-1)*(n-2): n in [2..40]]; // Vincenzo Librandi, May 02 2011

Drop[CoefficientList[Series[6 x^3*(3 + x)/(1 - x)^5, {x, 0, 34}], x], 2] (* Michael De Vlieger, May 21 2021 *)

a(n)=n^4 - 3*n^3 + 2*n^2 \\ Charles R Greathouse IV, May 02 2011

a(n) = A004320(n-2)*6.
G.f.: 6*x^3*(3 + x)/(1 - x)^5. - Stefano Spezia, May 20 2021
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, May 22 2021
From Amiram Eldar, May 25 2021: (Start)
Sum_{n>=3} 1/a(n) = (Pi^2 - 9)/12.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 + 2*log(2) - 7/4. (End)

A098916 Permanent of the n X n (0,1)-matrices with ij-th entry equal to zero iff (i=1,j=1),(i=1,j=n),(i=n,j=1) and (i=n,j=n).

Original entry on oeis.org

0, 4, 36, 288, 2400, 21600, 211680, 2257920, 26127360, 326592000, 4390848000, 63228211200, 971415244800, 15866448998400, 274611617280000, 5021469573120000, 96746980442112000, 1959126353952768000

Offset: 3

Simone Severini, Oct 17 2004

nonn

The number of all possible ways to permute n distinct aligned balls, one is blue, 2 are red and the remaining are green, such that no red ball occurs by the side of the blue ball. It may generalized to r red balls: a(n,r) = (n-r-1)(n-r)(n-2)!. - Alessandro Nicolosi (xxalenicxx(AT)hotmail.com), Jul 12 2006
A formula for the permanents of these n X n matrices(A) can be easily derived by minor expansion along the first row: a(n)=per(A)=(n-2)*per(B), where B is the n-1 X n-1 (0,1)-matrix with bij=0 iff (i=n,j=1) and (i=n,j=n). A new minor expansion along the last row of B yields: per(B)=(n-3)*per(C)=(n-3)*(n-2)! since C is the n-2 X n-2 1-matrix. Hence: a(n)=(n-2)*(n-3)*(n-2)!. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007
Number of permutations of n-1 having exactly 4 points P on the boundary of their bounding square. (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 0David Nacin, Feb 27 2012
a(n) is also the number of permutations of n symbols that 4-commute with a transposition (see A233440 for definition): a permutation p of {1,...,n} has exactly four points on the boundary of their bounding square if and only if p 4-commutes with transposition (1, n). - Luis Manuel Rivera Martínez, Feb 27 2014

			a(3) = 0 because no configuration is allowed, the 2 red balls always occurs by the side of the blue ball. a(4) = 4 because we can have 4 possible permutations: b,g1,r1,r2 b,g1,r2,r1 r1,r2,g1,b r2,r1,g1,b.

Vincenzo Librandi, Table of n, a(n) for n = 3..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. A208528, A208529.

Cf. A001113, A001620, A099285.

a:= n->(n-2)*(n-3)*(n-2)!: seq(a(n), n=3..20); # Zerinvary Lajos, Jul 01 2007

a[n_,r_] := (n-r-1)(n-r)(n-2)! (* Alessandro Nicolosi (xxalenicxx(AT)hotmail.com), Jul 12 2006 *)
Table[(n-2)*(n-3)*(n-2)!,{n,3,30}] (* Vincenzo Librandi, Feb 27 2012 *)

permRWNb(a)=n=matsize(a)[1]; if(n==1,return(a[1,1])); sg=1; in=vectorv(n); x=in; x=a[,n]-sum(j=1,n,a[,j])/2; p=prod(i=1,n,x[i]); for(k=1,2^(n-1)-1,sg=-sg; j=valuation(k,2)+1; z=1-2*in[j]; in[j]+=z; x+=z*a[,j]; p+=prod(i=1,n,x[i],sg)); return(2*(2*(n%2)-1)*p)
for(n=3,24,a=matrix(n,n,i,j,1); a[1,1]=0; a[1,n]=0; a[n,1]=0; a[n,n]=0; print1(permRWNb(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

for(n=3,24,print1((n-2)*(n-3)*(n-2)!", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

import math
def a(n):
    return (n-2)*(n-3)*math.factorial(n-2) # David Nacin, Feb 27 2012

a(n) = (n-2)*(n-3)*(n-2)!. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=4} 1/a(n) = 3 - e, where e = A001113.
Sum_{n>=4} (-1)^n/a(n) = 2*(gamma - Ei(-1)) - 1/e - 1, where gamma = A001620 and Ei(-1) = -A099285. (End)

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

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)

A027764 a(n) = (n+1)*binomial(n+1,4).

Original entry on oeis.org

4, 25, 90, 245, 560, 1134, 2100, 3630, 5940, 9295, 14014, 20475, 29120, 40460, 55080, 73644, 96900, 125685, 160930, 203665, 255024, 316250, 388700, 473850, 573300, 688779, 822150, 975415, 1150720, 1350360, 1576784, 1832600, 2120580, 2443665, 2804970, 3207789

Offset: 3

Thi Ngoc Dinh (via R. K. Guy)

Number of 6-subsequences of [ 1, n ] with just 1 contiguous pair.

a(n) is also the number of permutations of n+1 symbols that 4-commute with an (n+1)-cycle (see A233440 for definition). - Luis Manuel Rivera Martínez, Feb 07 2014

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A233440.

[(n+1)*Binomial(n+1,4): n in [3..35]]; // Vincenzo Librandi, Feb 08 2014

Table[(n + 1)Binomial[n + 1, 4], {n, 3, 40}] (* or *) LinearRecurrence[ {6, -15, 20, -15, 6, -1}, {4, 25, 90, 245, 560, 1134}, 40] (* Harvey P. Dale, Jun 14 2013 *)
CoefficientList[Series[(4 + x)/(1 - x)^6, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 08 2014 *)

G.f.: (4+x)*x^3/(1-x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Jun 14 2013
a(n) = 10*C(n+2,2)*C(n+2,5)/(n+2)^2. - Gary Detlefs, Aug 20 2013
Sum_{n>=3} 1/a(n) = 62/9 - (2/3)*Pi^2. - Jaume Oliver Lafont, Jul 15 2017
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/3 + 80*log(2)/3 - 194/9. - Amiram Eldar, Jan 28 2022

A027765 a(n) = (n+1)*binomial(n+1,5).

Original entry on oeis.org

5, 36, 147, 448, 1134, 2520, 5082, 9504, 16731, 28028, 45045, 69888, 105196, 154224, 220932, 310080, 427329, 579348, 773927, 1020096, 1328250, 1710280, 2179710, 2751840, 3443895, 4275180, 5267241, 6444032, 7832088, 9460704, 11362120, 13571712, 16128189

Offset: 4

Thi Ngoc Dinh (via R. K. Guy)

Number of 7-subsequences of [ 1, n ] with just 1 contiguous pair.

8*a(n) is the number of permutations of (n+1) symbols that 5-commute with an (n+1)-cycle (see A233440 for definition), where 8 = A000757(5). - Luis Manuel Rivera Martínez, Feb 07 2014

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000757, A233440.

[(n+1)*Binomial(n+1,5): n in [4..40]]; // Vincenzo Librandi, Aug 09 2017

a:=n->(sum((numbcomp(n,6)), j=2..n)):seq(a(n), n=6..34); # Zerinvary Lajos, Aug 26 2008

Table[(n+1)Binomial[n+1,5],{n,4,40}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{5,36,147,448,1134,2520,5082},40] (* Harvey P. Dale, Jan 15 2017 *)

G.f.: (5+x)*x^4/(1-x)^7.
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=4} 1/a(n) = 5*Pi^2/6 - 575/72.
Sum_{n>=4} (-1)^n/a(n) = 5*Pi^2/12 + 160*log(2)/3 - 2945/72. (End)

Incorrect formula deleted by R. J. Mathar, Feb 13 2016

A027769 a(n) = (n+1)*binomial(n+1, 9).

Original entry on oeis.org

9, 100, 605, 2640, 9295, 28028, 75075, 183040, 413270, 875160, 1755182, 3359200, 6172530, 10943240, 18795370, 31380096, 51074375, 81238300, 126544275, 193393200, 290435145, 429214500, 624962325, 897561600, 1272714300, 1783342704, 2471261100, 3389158080

Offset: 8

Thi Ngoc Dinh (via R. K. Guy)

nonn

Number of 11-subsequences of [ 1, n ] with just 1 contiguous pair.

13208*a(n) is the number of permutations of (n+1) symbols that 9-commute with an (n+1)-cycle (see A233440 for definition), where 13208=A000757(9). - Luis Manuel Rivera Martínez, Feb 06 2014

T. D. Noe, Table of n, a(n) for n = 8..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000757, A233440.

Table[(n+1)*Binomial[n+1, 9], {n, 8, 35}] (* Amiram Eldar, Jan 30 2022 *)

G.f.: (9+x)*x^8/(1-x)^11.
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=8} 1/a(n) = 3*Pi^2/2 - 575499/39200.
Sum_{n>=8} (-1)^n/a(n) = 3*Pi^2/4 + 24576*log(2)/35 - 19365109/39200. (End)

A027770 a(n) = (n + 1)*binomial(n + 1, 10).

Original entry on oeis.org

10, 121, 792, 3718, 14014, 45045, 128128, 330616, 787644, 1755182, 3695120, 7407036, 14226212, 26313518, 47070144, 81719000, 138105110, 227779695, 367447080, 580870290, 901350450, 1374917115, 2064391680, 3054514320, 4458356760, 6425278860, 9150726816

Offset: 9

Thi Ngoc Dinh (via R. K. Guy)

Number of 12-subsequences of [ 1, n ] with just one contiguous pair.

120288*a(n) is the number of permutations of (n+1) symbols that 10-commute with an (n+1)-cycle (see A233440 for definition), where 120288 = A000757(10). - Luis Manuel Rivera Martínez, Feb 07 2014

T. D. Noe, Table of n, a(n) for n = 9..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000757, A233440.

a:= n-> (n+1)*binomial(n+1, 10):
seq(a(n), n=9..36);  # Alois P. Heinz, Oct 04 2019

((# + 1) Binomial[# + 1, 10] &) /@ Range[9, 48] (* Alonso del Arte, Oct 04 2019 *)

G.f.: (10 + x)*x^9/(1 - x)^12.
a(n) = C(n + 1, 10)*C(n + 1, 1). - Zerinvary Lajos, Jun 08 2005, corrected by R. J. Mathar, Feb 13 2016
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=9} 1/a(n) = 5257891/317520 - 5*Pi^2/3.
Sum_{n>=9} (-1)^(n+1)/a(n) = 5*Pi^2/6 + 84992*log(2)/63 - 299498341/317520. (End)

A027771 a(n) = (n+1)*binomial(n+1,11).

Original entry on oeis.org

11, 144, 1014, 5096, 20475, 69888, 210392, 572832, 1436058, 3359200, 7407036, 15519504, 31097794, 59907456, 111435000, 200880160, 352023165, 601277040, 1003321410, 1638819000, 2624841765, 4128783360, 6386711760, 9727323840, 14602906500, 21628990656

Offset: 10

Thi Ngoc Dinh (via R. K. Guy)

Number of 13-subsequences of [ 1, n ] with just 1 contiguous pair.

1214673*a(n) is the number of permutations of (n+1) symbols that 11-commute with an (n+1)-cycle (see A233440 for definition), where 1214673 = A000757(11). - Luis Manuel Rivera Martínez, Feb 07 2014

T. D. Noe, Table of n, a(n) for n = 10..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A000757, A233440.

Table[(n+1)*Binomial[n+1, 11], {n, 10, 35}] (* Amiram Eldar, Jan 30 2022 *)

G.f.: (11+x)*x^10/(1-x)^13.
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=10} 1/a(n) = 11*Pi^2/6 - 57138257/3175200.
Sum_{n>=10} (-1)^n/a(n) = 11*Pi^2/12 + 822272*log(2)/315 - 5773608863/3175200. (End)

A027766 a(n) = (n+1)*binomial(n+1,6).

Original entry on oeis.org

6, 49, 224, 756, 2100, 5082, 11088, 22308, 42042, 75075, 128128, 210392, 334152, 515508, 775200, 1139544, 1641486, 2321781, 3230304, 4427500, 5985980, 7992270, 10548720, 13775580, 17813250, 22824711, 28998144, 36549744, 45726736, 56810600, 70120512, 86017008

Offset: 5

Thi Ngoc Dinh (via R. K. Guy)

Number of 8-subsequences of [ 1, n ] with just 1 contiguous pair.

36*a(n) is the number of permutations of (n+1) symbols that 6-commute with an (n+1)-cycle (see A233440 for definition), where 36 = A000757(6). - Luis Manuel Rivera Martínez, Feb 07 2014

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000757, A233440.

G.f.: (6+x)*x^5/(1-x)^8.
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=5} 1/a(n) = 3019/300 - Pi^2.
Sum_{n>=5} (-1)^(n+1)/a(n) = Pi^2/2 + 512*log(2)/5 - 22729/300. (End)

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A004320 a(n) = n*(n+1)*(n+2)^2/6.

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A047929 a(n) = n^2*(n-1)*(n-2).

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A098916 Permanent of the n X n (0,1)-matrices with ij-th entry equal to zero iff (i=1,j=1),(i=1,j=n),(i=n,j=1) and (i=n,j=n).

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PARI

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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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A027764 a(n) = (n+1)*binomial(n+1,4).

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A027765 a(n) = (n+1)*binomial(n+1,5).

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A027769 a(n) = (n+1)*binomial(n+1, 9).

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A004320 a(n) = n(n+1)(n+2)^2/6.

A047929 a(n) = n^2(n-1)(n-2).