A001286 -id:A001286 - cp's OEIS frontend

A385366 a(n) = Sum_{permutations p of [n]} des(p^2), where des(p) is the number of descents of p.

0, 0, 2, 24, 192, 1560, 13680, 131040, 1370880, 15603840, 192326400, 2554675200, 36404121600, 554204851200, 8979363993600, 154305575424000, 2803653844992000, 53708801642496000, 1082001156268032000, 22869278876860416000, 506043617700741120000, 11699825757321461760000

Offset: 1

Yifan Xie, Jun 26 2025

			For the permutation p = (2, 3, 4, 1), p^2 = (3, 4, 1, 2), and des(p) = des(p^2) = 1 (because 4 > 1).

Paolo Xausa, Table of n, a(n) for n = 1..445

Cf. A001286.

A385366[n_] := If[n <= 2, 0, (n - 1)!*(n^2 - n - 4)/2];
Array[A385366, 25] (* Paolo Xausa, Jul 14 2025 *)

PARI
```
a(n)=if(n>2,(n-1)!*(n^2-n-4)/2, 0);
```

a(n) = 0 if n <= 2; a(n) = (n-1)!*(n^2-n-4)/2 if n >= 3.

A060617 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=7 and D varies.

Original entry on oeis.org

0, 1, 18, 9600

Offset: 7

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A060618 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=8 and D varies.

Original entry on oeis.org

0, 1, 20, 22528

Offset: 8

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).

Original entry on oeis.org

0, 1, 0, 1, -1, -1, -1, 0, 0, 3, 1, -5, 18, -13, -7, -11, 70, -135, 65, -10, 45, 111, -609, 1215, -1350, 1275, -621, -141, -1009, 6188, -16758, 27335, -26845, 12474, -2548, 1883, 10977, -81353, 270004, -511791, 584710, -420287, 216468, -70169, -3599, -146691, 1248210, -4715217, 10303461, -14439411

Offset: 0

Gerald McGarvey, Mar 02 2005

For n>0 the row sums = (-1)^(n-1) * (n-1)! For n odd, the sum of the absolute values of the coefficients in the n-th row = (2*(n-1))!/n! (every other entry of A001761).
The sum of the (2n)th du Bois-Reymond constants = 1/5 or is very close to 1/5.
For the 6th and 9th rows, the coefficients were adjusted from results of the residue evaluations so that double factorials ((2n)!! = 2^n*n! (A000165)) are in the denominators. For the 6th row they were multiplied by 3, for the 9th row they were multiplied by 9.
For n>1, Sum_{k=0..n} (n-k+1)*a(n,k) = (-1)^(n)*A001286(n-1) [A001286 are Lah numbers: (n-1)*n!/2].

Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, du Bois-Reymond Constants.

Cf. A000142, A000165, A001761, A062545, A062546, A085466, A085467.

Table[2 Residue[x^2/((1+x^2)^n (Tan[x]-x)), {x, I}], {n, 0, 9}]

For n>1, C_2n = -3 - 2 * Residue_{x=i} (x^2/((1+x^2)^n * (tan(x) - x))) (see MathWorld article).

For n>1, Sum_{k=0..n} (-1)^(n+k)*a(n, k) = (2*(n-1))!/n! (i.e., A001761(n-1)).

Added the keyword tabl Gerald McGarvey, Aug 20 2009

A105187 a(n) = determinant of the n X n matrix m(i,j) = (i+j+2)!/i!/j!.

Original entry on oeis.org

1, 24, 720, 28800, 1512000, 101606400, 8534937600, 877879296000, 108637562880000, 15933509222400000, 2734190182563840000, 542861032610856960000, 123500884918969958400000, 31920228717518389248000000, 9302466654819644866560000000, 3036325116133132084445184000000

Offset: 0

Benoit Cloitre, Apr 11 2005

nonn

Cf. A001286.

PARI
```
a(n)=(1/12)*(n+1)*(n+2)!*(n+3)!
```

a(n) = (1/12)*(n+1)*(n+2)!*(n+3)!.

A105188 a(n) = determinant of the n X n matrix m(i,j)=(i+j+3)!/i!/j!.

Original entry on oeis.org

1, 120, 21600, 6048000, 2540160000, 1536288768000, 1290482565120000, 1460088845107200000, 2168231934984192000000, 4134095556036526080000000, 9931751163822150254592000000, 29578560738801240212766720000000

Offset: 0

Benoit Cloitre, Apr 11 2005

nonn

Cf. A001286.

PARI
```
a(n)=(1/288)*(n+1)*(n+2)!*(n+3)!*(n+4)!
```

a(n)=(1/288)*(n+1)*(n+2)!*(n+3)!*(n+4)!

A141052 Number of runs or rising sequences of length 2 among all permutations of n.

Original entry on oeis.org

1, 4, 21, 130, 930, 7560, 68880, 695520, 7711200, 93139200, 1217462400, 17124307200, 257902444800, 4140968832000, 70614415872000, 1274546617344000, 24275666967552000, 486580401635328000, 10238462617743360000, 225651661258383360000, 5198503365971435520000

Offset: 2

Harlan J. Brothers, Jul 31 2008, Aug 24 2008

			a[3]=4 because of the 6 permutations of n=3, there are 4 ascending runs of length 2:
{1,3} in {1,3,2}
{1,3} in {2,1,3}
{2,3} in {2,3,1}
{1,2} in {3,1,2}
a[3]=4 because of the 6 permutations of n=3, there are 4 rising sequences of length 2:
{1,2} in {1,3,2}
{2,3} in {2,1,3}
{2,3} in {2,3,1}
{1,2} in {3,1,2}

Persi Diaconis, Mathematical developments from the analysis of riffle shuffling, p. 4.
Francis Edward Su, Rising Sequences in Card Shuffling
Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society, 1997, pp.120-131.

Column 2 of A122843.

Cf. A008292, A097900, A001286, A001048, A000142, A028387, A001710.

Table[n!(5n + 1)/4! + Floor[2/n](1/12), {n, 2, 10}]

a(n) = n!*(5n+1)/4! + floor(2/n)*(1/12), n>=2.
Recurrence: a(n) = (n+1)*a(n-1)+(n-1)!/6, n>=2, with a(2)=1 and a(3)=4.
E.g.f.: x^2*(x-2)*(x-6)/(24*(x-1)^2).

First example and typo in second example corrected by Harlan J. Brothers, Apr 29 2013

A181416 Irregular table T(n,k) = n*A178883(n,k) read by rows.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 24, 32, 16, 72, 96, 120, 120, 120, 180, 180, 480, 600, 720, 576, 576, 288, 648, 1296, 216, 1152, 1728, 3600, 4320, 5040, 3360, 3360, 3360, 3024, 6048, 3024, 3024, 4032, 12096, 4032, 8400, 16800, 30240, 35280, 40320

Offset: 1

Alford Arnold, Oct 17 2010

The row sum of row n is A001286(n).

			In row n=3 the products are (3,3,3) times (2,4,6) yielding (6,12,18) which adds to 36, the third Lah number.
The table starts in row n=1 with row lengths A000041(n) as:
1;
2,4;
6,12,18;
24,32,16,72,96;
120,120,120,180,180,480,600;

Cf. A051683.

T(n,k) = A036042(n,k)*A178883(n,k), 1<=k<= A000041(n).

A281485 Triangular array T(n,k) = k Sum_{j=0..k-1} (-1)^j binomial(k-1,j) (n-1-j)^(n-1), 1<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 4, 6, 6, 27, 38, 36, 24, 256, 350, 330, 240, 120, 3125, 4202, 3960, 3000, 1800, 720, 46656, 62062, 58506, 45360, 29400, 15120, 5040, 823543, 1087214, 1025388, 806904, 546000, 312480, 141120, 40320, 16777216, 22024830, 20781690, 16524144, 11493720, 6985440, 3598560, 1451520, 362880

Offset: 1

Rui Duarte, Jan 22 2017

A parking function of size n is a sequence (a_1,...,a_n) of positive integers such that, if b_1 <= b_2 <= ... <= b_n is the increasing rearrangement of the sequence (a_1,..,a_n), then b_i <= i.
Given a:[n]->[n], the center of a is the largest subset Z(a) = { z_1, ..., z_k } of [n] such that z_1 < z_2 < ... < z_k and a_(z_j) <= j, for every j in [k]. The length of the center of a is |Z(a)|.
Then T(n,k)= number of parking functions of size n with center of length k.

			First seven rows:
      1
      1      2
      4      6      6
     27     38     36     24
    256    350    330    240    120
   3125   4202   3960   3000   1800    720
  46656  62062  58506  45360  29400  15120   5040

Rui Duarte and António Guedes de Oliveira, The number of parking functions with center of a given length, arXiv:1611.03707 (2016).

T(n,k) = k * A174551(n-1,k-1).
T(n,1) = (n-1)^(n-1) = A000312(n-1).
T(n,n-1) = n!(n-1)/2 = A001286(n), n>=2.
T(n,n) = n! = A000142(n).
Sum_{i=1,...,n} T(n,i) = (n+1)^(n-1) = A000272(n+1).

Table[Which[n == k == 1, 1, k == 1, (n - 1)^(n - 1), k == n, n!, True, k Sum[(-1)^j*Binomial[k - 1, j] (n - 1 - j)^(n - 1), {j, 0, k - 1}]], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 23 2017 *)

T(n,k) = k*Sum_{j=0..k-1} (-1)^j*binomial(k-1,j)*(n-1-j)^(n-1).

T(n,k) = k!*Sum_{j_1+j_2+...+j_k=n-k} (n-1)^(j_1)*(n-2)^(j_2)*...*(n-k)^(j_k).

A305739 a(n) = n!*T(n) - 1, where T(n) is the n-th triangular number.

Original entry on oeis.org

0, 5, 35, 239, 1799, 15119, 141119, 1451519, 16329599, 199583999, 2634508799, 37362124799, 566658892799, 9153720575999, 156920924159999, 2845499424767999, 54420176498687999, 1094805903679487999, 23112569077678079999, 510909421717094399999

Offset: 1

Maheswara Rao Valluri, Jun 22 2018

nonn

Cf. A000142, A000217, A001286, A180119.

See A305738 for the indices of primes in this sequence.

Maple
```
seq(n*(n+1)!/2-1,n=1..21);
```

a(n) = n*(n+1)!/2 - 1; \\ Michel Marcus, Jun 23 2018

a(n) = n*(n+1)!/2 - 1. - Michel Marcus, Jun 23 2018

a(n) = A180119(n)-1 = A001286(n+1)-1. - Alois P. Heinz, Jun 24 2018

cp's OEIS Frontend

A385366 a(n) = Sum_{permutations p of [n]} des(p^2), where des(p) is the number of descents of p.

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A060617 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=7 and D varies.

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A060618 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=8 and D varies.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)*tanh(1)^k) / (2^n*n! * (1-tanh(1))^n).

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A105187 a(n) = determinant of the n X n matrix m(i,j) = (i+j+2)!/i!/j!.

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PARI

Formula

A105188 a(n) = determinant of the n X n matrix m(i,j)=(i+j+3)!/i!/j!.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A141052 Number of runs or rising sequences of length 2 among all permutations of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A181416 Irregular table T(n,k) = n*A178883(n,k) read by rows.

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Author

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Comments

Examples

Crossrefs

Formula

A281485 Triangular array T(n,k) = k Sum_{j=0..k-1} (-1)^j binomial(k-1,j) (n-1-j)^(n-1), 1<=k<=n, read by rows.

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A104053 Triangle of coefficients in the numerators of rational functions in tanh(1) that express the (2n)th du Bois-Reymond constants as C_0 = 0, C_2 = -4 - 1/(1-tanh(1)), for n>1, C_2n = -3 - (Sum_{k=0..n} a(n,k)tanh(1)^k) / (2^nn! * (1-tanh(1))^n).