A110468 -id:A110468 - cp's OEIS frontend

A145877 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the shortest cycle length is k (1<=k<=n).

1, 1, 1, 4, 0, 2, 15, 3, 0, 6, 76, 20, 0, 0, 24, 455, 105, 40, 0, 0, 120, 3186, 714, 420, 0, 0, 0, 720, 25487, 5845, 2688, 1260, 0, 0, 0, 5040, 229384, 52632, 22400, 18144, 0, 0, 0, 0, 40320, 2293839, 525105, 223200, 151200, 72576, 0, 0, 0, 0, 362880, 25232230

Offset: 1

Emeric Deutsch, Oct 27 2008

Row sums are the factorials (A000142).
Sum(T(n,k), k=2..n) = A000166(n) (the derangement numbers).
T(n,1) = A002467(n).
T(n,n) = (n-1)! (A000142).
Sum(k*T(n,k),k=1..n) = A028417(n).
For the statistic "length of the longest cycle", see A126074.

			T(4,2)=3 because we have 3412=(13)(24), 2143=(12)(34) and 4321=(14)(23).
Triangle starts:
      1;
      1,    1;
      4,    0,    2;
     15,    3,    0,    6;
     76,   20,    0,    0, 24;
    455,  105,   40,    0,  0, 120;
   3186,  714,  420,    0,  0,   0, 720;
  25487, 5845, 2688, 1260,  0,   0,   0, 5040;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Cf. A000142, A000166, A002467, A028417, A126074.

T(2n,n) gives A110468(n-1) (for n>0). - Alois P. Heinz, Apr 21 2017

F:=proc(k) options operator, arrow: (1-exp(-x^k/k))*exp(-(sum(x^j/j, j = 1 .. k-1)))/(1-x) end proc: for k to 16 do g[k]:= series(F(k),x=0,16) end do: T:= proc(n,k) options operator, arrow: factorial(n)*coeff(g[k],x,n) end proc: for n to 11 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

Rest[Transpose[Table[Range[0, 16]! CoefficientList[
      Series[(Exp[x^n/n] -1) (Exp[-Sum[x^k/k, {k, 1, n}]]/(1 - x)), {x, 0, 16}],x], {n, 1, 8}]]] // Grid (* Geoffrey Critzer, Mar 04 2011 *)

E.g.f. for column k is (1-exp(-x^k/k))*exp( -sum(j=1..k-1, x^j/j ) ) / (1-x). - Vladeta Jovovic

A094310 Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

Original entry on oeis.org

1, 2, 1, 6, 3, 2, 24, 12, 8, 6, 120, 60, 40, 30, 24, 720, 360, 240, 180, 144, 120, 5040, 2520, 1680, 1260, 1008, 840, 720, 40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040, 362880, 181440, 120960, 90720, 72576, 60480, 51840, 45360, 40320, 3628800, 1814400, 1209600, 907200, 725760, 604800, 518400, 453600, 403200, 362880

Offset: 1

Amarnath Murthy, Apr 29 2004

The sum of the rows gives A000254 (Stirling numbers of first kind). The first column and the leading diagonal are factorials given by A000142 with offsets of 0 and 1.
T(n,k) is the number of length k cycles in all permutations of {1..n}.
Second diagonal gives A001048(n). - Anton Zakharov, Oct 24 2016
T(n,k) is the number of permutations of [n] with all elements of [k] in a single cycle. To prove this result, let m denote the length of the cycle containing {1,..,k}. Letting m run from k to n, we obtain T(n,k) = Sum_{m=k..n} (C(n-k,m-k)*(m-1)!*(n-m)!) = n!/k. See an example below. - Dennis P. Walsh, May 24 2020

			Triangle begins as:
      1;
      2,     1;
      6,     3,     2;
     24,    12,     8,     6;
    120,    60,    40,    30,   24;
    720,   360,   240,   180,  144,  120;
   5040,  2520,  1680,  1260, 1008,  840,  720;
  40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040;
  ...
T(4,2) counts the 12 permutations of [4] with elements 1 and 2 in the same cycle, namely, (1 2)(3 4), (1 2)(3)(4), (1 2 3)(4), (1 3 2)(4), (1 2 4)(3), (1 4 2)(3), (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), and (1 4 3 2). - _Dennis P. Walsh_, May 24 2020

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A000142, A000254, A001563, A001710, A002301.
Cf. A061579, A094307, A110468, A133799.
Cf. A129825. - Johannes W. Meijer, Jun 18 2009

Maple
```
seq(seq(n!/k, k=1..n), n=1..10);
```

Table[n!/k, {n,10}, {k,n}]//Flatten
Table[n!/Range[n], {n,10}]//Flatten (* Harvey P. Dale, Mar 12 2016 *)

E.g.f. for column k: x^k/(k*(1-x)).

T(n,k)*k = n*n! = A001563(n).

More terms from Philippe Deléham, Jun 11 2005

A349979 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-longest cycle has length exactly k; n>=0, 0<=k<=floor(n/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 6, 15, 3, 24, 61, 35, 120, 290, 270, 40, 720, 1646, 1974, 700, 5040, 11025, 14707, 8288, 1260, 40320, 85345, 117459, 90272, 29484, 362880, 749194, 1023390, 974720, 446040, 72576, 3628800, 7347374, 9813210, 10666480, 6332040, 2128896

Offset: 0

Steven Finch, Dec 07 2021

If the permutation has no second cycle, then its second-longest cycle is defined to have length 0.

			Triangle begins:
[0]     1;
[1]     1;
[2]     1,     1;
[3]     2,     4;
[4]     6,    15,      3;
[5]    24,    61,     35;
[6]   120,   290,    270,    40;
[7]   720,  1646,   1974,   700;
[8]  5040, 11025,  14707,  8288,  1260;
[9] 40320, 85345, 117459, 90272, 29484;
    ...

Alois P. Heinz, Rows n = 0..200, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Column 0 gives 1 together with A000142.
Column 1 gives 1 - (n-1)! + A006231(n).
Row sums give A000142.
T(2n,n) gives A110468(n-1) for n>=1.
Cf. A126074, A145877, A332851, A349980, A350015, A350016, A350273, A350274.

b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*
      b(n-j, sort([l[], j])[2..3])*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n, [0$2])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 07 2021

b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[(j - 1)!*b[n - j, Sort[ Append[l, j]][[2 ;; 3]]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := With[{p = b[n, {0, 0}]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Sum_{k=0..floor(n/2)} k * T(n,k) = A332851(n). - Alois P. Heinz, Dec 07 2021

A290053 Triangle read by rows: Polynomial coefficients per comment.

Original entry on oeis.org

1, 1, 0, 1, -2, 3, 1, -5, 10, 0, 1, -9, 31, -39, 40, 1, -14, 77, -196, 252, 0, 1, -20, 162, -664, 1457, -1476, 1260, 1, -27, 303, -1809, 6168, -11772, 12176, 0, 1, -35, 520, -4250, 20773, -61595, 107730, -95400, 72576, 1, -44, 836, -8954, 59279, -249986

Offset: 1

Gregory Gerard Wojnar, Jul 19 2017

Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See arXiv:1706.08381 [math,GM], 2017.] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N) = ((-1)^D/(D-1)!)(D-N)N^chi*g_D(N) where chi = (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). Then a(n) are the coefficients of the polynomials N^chi*g_D(N), starting at D=2. The leading term of each row is 1 (polynomials are monic). The final terms in all even rows are 0. In each row, terms alternate in sign.

			Triangle begins:
1;
1,   0;
1,  -2,   3;
1,  -5,  10,     0;
1,  -9,  31,   -39,    40;
1, -14,  77,  -196,   252,      0;
1, -20, 162,  -664,  1457,  -1476,   1260;
1, -27, 303, -1809,  6168, -11772,  12176,      0;
1, -35, 520, -4250, 20773, -61595, 107730, -95400, 72576;
...

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.
Gregory Gerard Wojnar, java_program which (1) creates Maple program to create polynomial referenced in Comment, and (2) creates list of polynomial portion's coefficients (without trailing 0 constant term is odd degree cases) which constitute the rows of this triangle. Each run of the program is for a single degree; to change the degree the user must modify the value of "level" in line 393 of the java code.

The final terms in odd-numbered rows are A110468.
The negation of the second column give A000096.
The 3rd column is A290061; negation of 4th column is A290071; 5th column is A290127. Up to sign, all columns are given by polynomials described in the comments and examples of triangle A290761.

A202768 Vandermonde determinant of the first n squares.

Original entry on oeis.org

1, 1, 3, 120, 151200, 10973491200, 73004442255360000, 64942882916646518784000000, 10615517921765466641283416064000000000, 419534029722194863260820186269027926016000000000000, 5103425917047830280023316797736216735574814664897331200000000000000

Offset: 0

Clark Kimberling, Jan 01 2012

nonn

Each term divides its successor, as in A110468.

a(m) is also the determinant of m X m matrix M(i,j) = i^(2*j)*cosh(2*j*arccsch(i)), with i from 1 to m, and j from 0 to m-1. - Federico Provvedi, Jan 20 2021

			a(3) = (4-1)(9-1)(9-4) = 120.

Alois P. Heinz, Table of n, a(n) for n = 0..29

Cf. A000290, A110468, A168467, A000142.

with(LinearAlgebra):
a:= n-> Determinant(VandermondeMatrix([i^2$i=1..n])):
seq(a(n), n=0..12);  # Alois P. Heinz, Aug 21 2014

f[j_] := j^2; z = 15;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]    (* A202768 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}] (* A110468 *)
(* or *)
Det@TrigExpand@Array[#1^(2*#2)*Cosh[2*#2*ArcCsch[#1]]&,{#,#},{1,0}]&/@Range@16 (* Federico Provvedi, Jan 20 2021 *)
Table[Exp[(n^2-1/24)*Log[2]-(n/2+1/4)*Log[Pi]+3/2*Log@Glaisher+Log@BarnesG[1+n]+Log@BarnesG[3/2+n]-1/8]/n!,{n, 0, 40}] (* Federico Provvedi, Apr 01 2021 after Vaclav Kotesovec's formula *)

a(n)=prod(k=1,n,(2*k-1)!/k) /* Paul D. Hanna, Jan 02 2012 */

from math import prod
def A202768(n): return (prod(((m:=k+1<<1)*(m+1))**(n-1-k)//(k+1) for k in range(1,n-1))*3**(n-1)<Chai Wah Wu, Nov 26 2023

a(n) = Product_{k=0..n-1} (2*k+1)!/(k+1) = Product_{k=0..n-1} A110468(k). - Paul D. Hanna, Jan 02 2012
a(n) ~ 2^(n^2 + n - 7/24) * n^(n^2 - n/2 - 13/24) * Pi^((n-1)/2) / (sqrt(A) * exp(3*n^2/2 - n/2 - 1/24)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 25 2019
a(n) = A168467(n) / A000142(n). - Federico Provvedi, Apr 01 2021
For n > 0, a(n) = sqrt(BarnesG(2*n)) * Gamma(2*n) / (n * Gamma(n)^(3/2) * 2^((n-1)/2)). - Vaclav Kotesovec, Nov 27 2024

a(0) from Alois P. Heinz, Aug 21 2014

A346085 Number T(n,k) of permutations of [n] such that k is the GCD of the cycle lengths; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 0, 2, 0, 15, 3, 0, 6, 0, 96, 0, 0, 0, 24, 0, 455, 105, 40, 0, 0, 120, 0, 4320, 0, 0, 0, 0, 0, 720, 0, 29295, 4725, 0, 1260, 0, 0, 0, 5040, 0, 300160, 0, 22400, 0, 0, 0, 0, 0, 40320, 0, 2663199, 530145, 0, 0, 72576, 0, 0, 0, 0, 362880

Offset: 0

Alois P. Heinz, Jul 04 2021

			T(3,1) = 4: (1)(23), (13)(2), (12)(3), (1)(2)(3).
T(4,4) = 6: (1234), (1243), (1324), (1342), (1423), (1432).
Triangle T(n,k) begins:
  1;
  0,       1;
  0,       1,      1;
  0,       4,      0,     2;
  0,      15,      3,     0,    6;
  0,      96,      0,     0,    0,    24;
  0,     455,    105,    40,    0,     0, 120;
  0,    4320,      0,     0,    0,     0,   0, 720;
  0,   29295,   4725,     0, 1260,     0,   0,   0, 5040;
  0,  300160,      0, 22400,    0,     0,   0,   0,    0, 40320;
  0, 2663199, 530145,     0,    0, 72576,   0,   0,    0,     0, 362880;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-1 give: A000007, A079128.
Even bisection of column k=2 gives A346086.
Row sums give A000142.
T(2n,n) gives A110468(n-1) for n >= 1.
Cf. A057731, A145877, A346066, A359951.

b:= proc(n, g) option remember; `if`(n=0, x^g, add((j-1)!
      *b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..12);

b[n_, g_] := b[n, g] = If[n == 0, x^g, Sum[(j - 1)!*
     b[n - j, GCD[g, j]] Binomial[n - 1, j - 1], {j, n}]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A346066(n).

Sum_{prime p <= n} T(n,p) = A359951(n). - Alois P. Heinz, Jan 20 2023

A203476 a(n) = v(n+1)/v(n), where v = A203475.

Original entry on oeis.org

5, 130, 8500, 1051076, 211255200, 62840245000, 25959932960000, 14224928867370000, 9986120745657472000, 8740787543400204500000, 9333385482079885824000000, 11942338721669302523305000000, 18038821394494464638896640000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..200

Cf. A093883, A110468, A203475.

[(&*[(n+1)^2 + j^2: j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023

(* First program *)
f[j_]:= j^2; z = 15;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]           (* A203475 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A203476 *)
(* Second program *)
Table[Product[j^2 +(n+1)^2 , {j,n}], {n,20}] (* G. C. Greubel, Aug 28 2023 *)

[product(j^2+(n+1)^2 for j in range(1,n+1)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023

a(n) ~ 2^(n + 1/2) * exp(Pi*(n+1)/2 - 2*n) * n^(2*n). - Vaclav Kotesovec, Jan 25 2019

a(n) = Product_{j=1..n} ((n+1)^2 + j^2). - G. C. Greubel, Aug 28 2023

A256881 a(n) = n!/ceiling(n/2).

Original entry on oeis.org

1, 2, 3, 12, 40, 240, 1260, 10080, 72576, 725760, 6652800, 79833600, 889574400, 12454041600, 163459296000, 2615348736000, 39520825344000, 711374856192000, 12164510040883200, 243290200817664000, 4644631106519040000, 102181884343418880000, 2154334728240414720000

Offset: 1

M. F. Hasler, Apr 22 2015

nonn

Original name was: n!/round(n/2). - Robert Israel, Sep 03 2018

Robert Israel, Table of n, a(n) for n = 1..450
Pierre-Alain Sallard, Sum of repeated integrals of sinh.

Cf. A009445, A052612, A052616, A052849, A081457, A208529, A098558, A107991, A110468, A229244, A256031.

[Factorial(n)/Round(n/2): n in [1..30]]; // Vincenzo Librandi, Apr 23 2015

Maple
```
A256881 := n!/round(n/2);
```

Function[x, 1/x] /@
CoefficientList[Series[(Sinh[x] + x*Exp[x])/2, {x, 0, 20}], x] (* Pierre-Alain Sallard, Dec 15 2018 *)

PARI
```
A256881(n)=n!/round(n/2)
```

a(2n) = 2*A009445(n) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A081457(3n-1) = A208529(2n+1) = A256031(2n-1).
a(2n+1) = A110468(n) = A107991(2n+2) = A229244(2n+1), n>=0.
From Robert Israel, Sep 03 2018: (Start)
E.g.f.: -(1+1/x)*log(1-x^2).
n*(n+1)*(n+2)*a(n)+(n+2)*a(n+1)-(n+3)*a(n+2)=0. (End)
a(n) = 2/([x^n](sinh(x) + x*exp(x))). - Pierre-Alain Sallard, Dec 15 2018
Sum_{n>=1} 1/a(n) = (3*e-1/e)/4 = (e + sinh(1))/2. - Amiram Eldar, Feb 02 2023

Definition clarified by Robert Israel, Sep 03 2018

A229244 Number of n-permutations such that at least one cycle has size ceiling(n/2).

Original entry on oeis.org

1, 1, 3, 9, 40, 200, 1260, 8820, 72576, 653184, 6652800, 73180800, 889574400, 11564467200, 163459296000, 2451889440000, 39520825344000, 671854030848000, 12164510040883200, 231125690776780800, 4644631106519040000, 97537253236899840000, 2154334728240414720000, 49549698749529538560000, 1193170003333152768000000

Offset: 1

Geoffrey Critzer, Sep 17 2013

nonn

			a(4) = 9 because we have:
1: (1)(2)(4,3)
2: (1)(3,2)(4)
3: (1)(4,2)(3)
4: (2,1)(3)(4)
5: (2,1)(4,3)
6: (3,1)(2)(4)
7: (3,1)(4,2)
8: (4,1)(2)(3)
9: (4,1)(3,2).

Cf. A110468.

f[n_]:=If[EvenQ[n],Binomial[n,n/2](n/2-1)!((n/2)!-(n/2-1)!)+n!/2/(n/2)^2,Binomial[n,Ceiling[n/2]]Floor[n/2]!^2]; Table[f[n],{n,1,25}]

For odd n,  a(2m+1)= binomial(2m+1,m+1)*m!^2.
For even n, a(2m) = binomial(2m,m)*(m-1)!*(m!-(m-1)!) + (2m)!/(2*m^2).
Conjecture: (n+1)*a(n) +(-3*n+1)*a(n-1) -(n-2)*(n^2-2*n-1)*a(n-2) +(n-2)*(n-3)^2*a(n-3)=0. - R. J. Mathar, May 23 2014

A273878 Numerator of (2*(n+1)!/(n+2)).

Original entry on oeis.org

1, 4, 3, 48, 40, 1440, 1260, 8960, 72576, 7257600, 6652800, 958003200, 889574400, 11623772160, 163459296000, 41845579776000, 39520825344000, 12804747411456000, 12164510040883200, 231704953159680000, 4644631106519040000

Offset: 0

Johannes W. Meijer, Jun 08 2016

The moments, i.e. E(X^n) = int(x^n * p(x), x = 0..infinity) for n > 0, of the probability density function p(x) = 2*x*E(x, 1, 1), see A163931, lead to this sequence.

			The first few moments of p(x) are: 1, 4/3, 3, 48/5, 40, 1440/7, … .

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

Cf. A090585 (denominators), A090586, A085250, A110468, A128059, A128060, A163931.

a := proc(n): numer(2*(n+1)!/(n+2)) end: seq(a(n), n=0..20);

a(n) = numerator(2*(n+1)!/(n+2)) \\ Felix Fröhlich, Jun 09 2016

a(n) = numer(2*(n+1)!/(n+2))
a(n) = (n+1) * A090586(n+1)
a(2*n) = A110468(n) and a(2*n+1) = (2*n)!*A085250(n+1)/A128060(n+2).

cp's OEIS Frontend

A145877 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the shortest cycle length is k (1<=k<=n).

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A094310 Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

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A349979 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-longest cycle has length exactly k; n>=0, 0<=k<=floor(n/2).

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A290053 Triangle read by rows: Polynomial coefficients per comment.

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A202768 Vandermonde determinant of the first n squares.

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A346085 Number T(n,k) of permutations of [n] such that k is the GCD of the cycle lengths; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A203476 a(n) = v(n+1)/v(n), where v = A203475.

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