A060706 -id:A060706 - cp's OEIS frontend

A052502 Number of permutations sigma of [3n] without fixed points such that sigma^3 = Id.

1, 2, 40, 2240, 246400, 44844800, 12197785600, 4635158528000, 2345390215168000, 1524503639859200000, 1237896955565670400000, 1227993779921145036800000, 1461312598106162593792000000, 2054605512937264606871552000000

Offset: 0

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

For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(3n) consisting of permutations whose cycle decomposition is a product of n disjoint 3-cycles.

F. W. J. Olver, Asymptotics and special functions, Academic Press, 1974, pages 336-344.

G. C. Greubel, Table of n, a(n) for n = 0..210
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 27
F. W. J. Olver et al., NIST Digital Library of Mathematical Functions, eq. 9.10.17.

Cf. A000142. Row sums of triangle A060063.
First column of array A091752 (also negative of second column).
Equals row sums of A157702. - Johannes W. Meijer, Mar 07 2009
Karol A. Penson suggested that the row sums of A060063 coincide with this entry.
Cf. A001147, A052504, A060706, A261317, A261381.
Trisection of column k=3 of A261430.

List([0..20], n-> Factorial(3*n)/(3^n*Factorial(n))) # G. C. Greubel, May 14 2019

[Factorial(3*n)/(3^n*Factorial(n)): n in [0..20]]; // G. C. Greubel, May 14 2019

spec := [S,{S=Set(Union(Cycle(Z,card=3)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[(3*n)!/(3^n*n!), {n, 0, 20}] (* G. C. Greubel, May 14 2019 *)

{a(n) = (3*n)!/(3^n*n!)}; \\ G. C. Greubel, May 14 2019

[factorial(3*n)/(3^n*factorial(n)) for n in (0..20)] # G. C. Greubel, May 14 2019

From Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001: (Start)
a(n) = (3*n)!/(3^n * n!).
a(n) ~ sqrt(3) * 9^n * (n/e)^(2n). (End)
E.g.f.: (every third coefficient of) exp(x^3/3).
G.f.: hypergeometric3F0([1/3, 2/3, 1], [], 9*x).
D-finite with recurrence a(n) = (3*n-1)*(3*n-2)*a(n-1) for n >= 1, with a(0) = 1.
Write the generating function for this sequence in the form A(x) = Sum_{n >= 0} a(n)* x^(2*n+1)/(2*n+1)!. The g.f. A(x) satisfies A'(x)*( 1 - A(x)^2) = 1. Robert Israel remarks that consequently A(x) is a root of z^3 - 3*z + 3*x with A(0) = 0. Cf. A001147, A052504 and A060706. - Peter Bala, Jan 02 2015
From Peter Bala, Feb 27 2024: (Start)
u(n) := a(n+1) satisfies the second-order recurrence u(n) = 18*n*u(n-1) + (3*n - 1)^2*(3*n - 2)^2*u(n-2) with u(0) = 2 and u(1) = 40.
A second solution to the recurrence is given by v(n) := u(n)*Sum_{k = 0..n} (-1)^k/((3*k + 1)*(3*k + 2)) with v(0) = 1 and v(1) = 18.
This leads to the continued fraction expansion (2/3)*log(2) = Sum_{k = 0..n} (-1)^k/((3*k + 1)*(3*k + 2)) = Limit_{n -> oo} v(n)/u(n) = 1/(2 + (1*2)^2/(18 + (4*5)^2/(2*18 + (7*8)^2/(3*18 + (10*11)^2/(4*18 +  ... ))))). (End)
From Gabriel B. Apolinario, Jul 30 2024: (Start)
a(n) = 3 * Integral_{t=0..oo} Ai(t)*t^(3*n) dt, where Ai(t) is the Airy function.
a(n) = Integral_{t=-oo..oo} Ai(t)*t^(3*n) dt. (End)

Edited by Wolfdieter Lang, Feb 13 2004

Title improved by Geoffrey Critzer, Aug 14 2015

A052504 Number of permutations sigma of [5n] without fixed points such that sigma^5 = Id.

Original entry on oeis.org

1, 24, 72576, 1743565824, 162193467211776, 41363226782215962624, 23578031983305871878782976, 26242915470187034742010543079424, 51804144968120491069562620291816882176, 168779147605615794796420686413626405734580224, 858246016274098851318874304509764200194078068965376

Offset: 0

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

For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(5n) consisting of permutations whose cycle decomposition is a product of n disjoint 5-cycles.

G. C. Greubel, Table of n, a(n) for n = 0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 29

Cf. A001147, A052502, A060706, A261317, A261381.

Quintisection of column k=5 of A261430.

List([0..15], n-> Factorial(5*n)/(5^n*Factorial(n))) # G. C. Greubel, May 14 2019

[Factorial(5*n)/(5^n*Factorial(n)): n in [0..15]]; // G. C. Greubel, May 14 2019

spec := [S,{S=Set(Union(Cycle(Z,card=5)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

nn = 50; Select[Range[0, nn]! CoefficientList[Series[Exp[x^5/5], {x, 0, nn}], x], # > 0 &]  (* Geoffrey Critzer, Aug 19 2012 *)

{a(n) = (5*n)!/(5^n*n!)}; \\ G. C. Greubel, May 14 2019

[factorial(5*n)/(5^n*factorial(n)) for n in (0..15)] # G. C. Greubel, May 14 2019

a(n) = (5n)! * [x^(5n)] exp(x^5/5).
From Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001: (Start)
a(n) = (5*n)! / (n! * 5^n).
a(0) = 1, a(1) = 24, for n >= 2 a(n) = a(n-1) * C(5*n - 1, 4)* 24 = (5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)*a(n-1).
a(n) ~ sqrt(5) * 625^n * (n/e)^(4n). (End)
Write the generating function for this sequence in the form A(x) = Sum_{n >= 0} a(n)* x^(4*n+1)/(4*n+1)!. Then A'(x)*( 1 - A(x)^4) = 1. Cf. A052502. - Peter Bala, Jan 02 2015

A067630 Denominators in power series for cos(x)*cosh(x).

Original entry on oeis.org

1, 6, 2520, 7484400, 81729648000, 2375880867360000, 151476660579404160000, 18608907752179801056000000, 4015057936610313875842560000000, 1419041926536183233139035980800000000, 778117449996850714059458989711872000000000

Offset: 0

Benoit Cloitre, Feb 02 2002

nonn

Cf. A000680, A060706.

f:= gfun:-rectoproc({a(n) - (64*n^4-96*n^3+44*n^2-6*n)*a(n-1), a(0)=1}, a(n), remember): map(f, [$0..20]); # Georg Fischer, Aug 17 2021

a[n_] := (4*n)!/4^n; Array[a, 10, 0] (* Amiram Eldar, Jan 18 2021 *)

my(x='x+O('x^50), v=apply(denominator, Vec(cos(x)*cosh(x)))); vector(#v\4, k, v[4*k-3]) \\ Michel Marcus, Jan 18 2021

cos(x)*cosh(x) = Sum_{n>=0} (-1)^n*x^(4*n)/a(n).
a(n) = (4*n)! / 4^n = A000680(2*n).
E.g.f.: 1/(1-x^4/4). - Mohammad K. Azarian, Mar 20 2012
a(n) = n!*A060706(n). - Bruno Berselli, Mar 21 2012
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=0} 1/a(n) = (cos(sqrt(2)) + cosh(sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = cos(1)*cosh(1). (End)
D-finite with recurrence: a(n) - (64*n^4 - 96*n^3 + 44*n^2 - 6*n)*a(n-1) = 0. - Georg Fischer, Aug 17 2021

A217360 a(n) = 2^nbinomial(4n, n)/(3*n+1).

Original entry on oeis.org

1, 2, 16, 176, 2240, 31008, 453376, 6888960, 107707392, 1721477120, 28000141312, 461964898304, 7712495058944, 130050777006080, 2211737871974400, 37892693797109760, 653389823437701120, 11330548232319664128, 197475886172892823552

Offset: 0

R. J. Mathar, Oct 01 2012

Old name was: Series reversion of x - 2*x^4.
Regular blocks of 2 intermediate zeros have been removed from the sequence: If y = x - 2*x^4, then x = y + 2*y^4 + 16*y^7 + 176*y^10 + 2240*y^13 + 31008*y^16 + ...
a(n) is the number of lattice paths (Schroeder paths) from (0,0) to (n,4n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 4x with the total number of occurrences of NE and D equal to n. - Michael D. Weiner, Jul 25 2019

Jinyuan Wang, Table of n, a(n) for n = 0..100
D. Birmajer, J. B. Gil, J. D. Gil and M. D. Weiner, Schröder Coloring and Applications, arXiv:1908.08103 [math.CO], 2019.

Cf. A153231 (x+2*x^3).

[2^n*Binomial(4*n, n)/(3*n+1): n in [0..25]]; // Vincenzo Librandi, Jul 26 2019

A100089 := proc(n)
    (3*n+1)! ;
end proc:
A060706 := proc(n)
    (4*n)!/n!/4^n ;
end proc:
A217360 := proc(n)
    8^(n)*A060706(n)/A100089(n) ;
end proc:
seq(A217360(n), n=0..20);

Table[2^n Binomial[4 n, n] / (3 n + 1), {n, 0, 20}] (* Vincenzo Librandi, Jul 26 2019 *)

D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n)- 8*(4*n-1)*(4*n-3)*(4*n-2)*a(n-1) = 0, so a(n) = 8^n*A060706(n)/A100089(n) = 2^n*A002293(n).
a(n) = [x^(3*n)](f(x)/x) where f(x) is the reversion of x - 2*x^4.
G.f.: F([1/4, 1/2, 3/4], [2/3, 4/3], 512*x/27), where F is the generalized hypergeometric function. - Stefano Spezia, Aug 18 2019
G.f. A(x) satisfies: A(x) = 1 / (1 - 2 * x * A(x)^3). - Ilya Gutkovskiy, Nov 12 2021

Offset decreased by 1 and name changed by Michael D. Weiner, Jul 25 2019

A319365 Expansion of e.g.f. exp(x^4/4)/(1 - x).

Original entry on oeis.org

1, 1, 2, 6, 30, 150, 900, 6300, 51660, 464940, 4649400, 51143400, 614968200, 7994586600, 111924212400, 1678863186000, 26865216378000, 456708678426000, 8220756211668000, 156194368021692000, 3123907159441068000, 65602050348262428000, 1443245107661773416000, 33194637476220788568000

Offset: 0

Ilya Gutkovskiy, Sep 17 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..449

Cf. A000138, A000522, A030975, A060706, A130905, A319364.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x^4/4)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Dec 28 2018

f:= gfun:-rectoproc({(n+1)*(n+2)*(n+3)*(n+4)*a(n)-(n+2)*(n+3)*(n+4)*a(n+1)-(n+5)*a(n+4)+a(n+5)},seq(a(i)=[1,1,2,6,30][i+1],i=0..4)},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 28 2018

nmax = 23; CoefficientList[Series[Exp[x^4/4]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) ~ n!*exp(1/4).

(n+1)*(n+2)*(n+3)*(n+4)*a(n)-(n+2)*(n+3)*(n+4)*a(n+1)-(n+5)*a(n+4)+a(n+5)=0. - Robert Israel, Dec 28 2018

A368213 Triangular array read by rows: Number of permutations of [n] that factor into exactly k-cycles, ordered by n (rows) and divisors k of n (columns).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 3, 0, 6, 1, 0, 0, 0, 24, 1, 15, 40, 0, 0, 120, 1, 0, 0, 0, 0, 0, 720, 1, 105, 0, 1260, 0, 0, 0, 5040, 1, 0, 2240, 0, 0, 0, 0, 0, 40320, 1, 945, 0, 0, 72576, 0, 0, 0, 0, 362880, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 1, 10395, 246400, 1247400, 0, 6652800, 0, 0, 0, 0, 0, 39916800

Offset: 1

Marko Riedel, Dec 17 2023

			Row n=6 is 1, 15, 40, 120 because there is one permutation of [6] consisting of six fixed points, there are 15 permutations consisting of three transpositions, there are forty permutations consisting of two three-cycles and there are one hundred and twenty permutations consisting of just one six-cycle (6!/6).
Triangular array starts:
[ 1] 1;
[ 2] 1,   1;
[ 3] 1,   0,    2;
[ 4] 1,   3,    0,    6;
[ 5] 1,   0,    0,    0,    24;
[ 6] 1,  15,   40,    0,     0, 120;
[ 7] 1,   0,    0,    0,     0,   0, 720;
[ 8] 1, 105,    0, 1260,     0,   0,   0, 5040;
[ 9] 1,   0, 2240,    0,     0,   0,   0,    0, 40320;
[10] 1, 945,    0,    0, 72576,   0,   0,    0,     0, 362880;

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, pages 120-122.

Marko Riedel, math.stackexchange.com, Number of permutations in the symmetric group.
Marko Riedel, Computing the number of permutations of [n] with k m-cycles with analytic combinatorics.
Marko Riedel, Computing the number of permutations of [n] with k m-cycles using the principle of inclusion-exclusion.

Cf. A005225 (row sums), A008290.

Cf. A123023 (column 2), A052502 (column 3), A060706 (column 4).

T:= (n, m)-> `if`(irem(n,m)=0, n!/m^(n/m)/(n/m)!, 0):
seq(seq(T(n, m), m = 1..n), n=1..15);

A368213[n_,k_]:=If[Divisible[n,k],n!/(k^(n/k)(n/k)!),0];
Table[A368213[n,k],{n,15},{k,n}] (* Paolo Xausa, Dec 18 2023 *)

def T(n, d): return factorial(n) // (d ** (n//d) * factorial(n//d))
for n in range(1, 19):
    print([T(n, d) if n % d == 0 else 0 for d in range(1, n+1)])
# Peter Luschny, Dec 17 2023

T(n, k) = n! / ( k^(n/k) * (n/k)! ) if k divides n otherwise 0.

A377597 Table read by antidiagonals: T(n,k) = (nk)!/(n^kk!), n >=1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 15, 40, 6, 1, 1, 105, 2240, 1260, 24, 1, 1, 945, 246400, 1247400, 72576, 120, 1, 1, 10395, 44844800, 3405402000, 1743565824, 6652800, 720, 1, 1, 135135, 12197785600, 19799007228000, 162193467211776, 4940103168000, 889574400, 5040, 1

Offset: 1

Peter Kagey, Nov 02 2024

This is the number of permutations in S_{k*n} that consist of k disjoint n-cycles.

			The table begins:
n\k| 0  1     2          3               4                    5
---+-----------------------------------------------------------
 1 | 1  1     1          1               1                    1
 2 | 1  1     3         15             105                  945
 3 | 1  2    40       2240          246400             44844800
 4 | 1  6  1260    1247400      3405402000       19799007228000
 5 | 1 24 72576 1743565824 162193467211776 41363226782215962624
For example T(2,5) = (2*5)!/(2^5*5!) = 10!/(32*5!) = 945.

Cf. A001147 (row 2), A052502 (row 3), A060706 (row 4), A052504 (row 5), A110468 (col 2).

Cf. A368213.

cp's OEIS Frontend

A052502 Number of permutations sigma of [3n] without fixed points such that sigma^3 = Id.

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A052504 Number of permutations sigma of [5n] without fixed points such that sigma^5 = Id.

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A067630 Denominators in power series for cos(x)*cosh(x).

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A217360 a(n) = 2^n*binomial(4*n, n)/(3*n+1).

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Magma

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A319365 Expansion of e.g.f. exp(x^4/4)/(1 - x).

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A368213 Triangular array read by rows: Number of permutations of [n] that factor into exactly k-cycles, ordered by n (rows) and divisors k of n (columns).

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A217360 a(n) = 2^nbinomial(4n, n)/(3*n+1).

A377597 Table read by antidiagonals: T(n,k) = (nk)!/(n^kk!), n >=1, k >= 0.