A261317 -id:A261317 - cp's OEIS frontend

A053496 Number of degree-n permutations of order dividing 6.

1, 1, 2, 6, 18, 66, 396, 2052, 12636, 91548, 625176, 4673736, 43575192, 377205336, 3624289488, 38829340656, 397695226896, 4338579616272, 54018173703456, 641634784488288, 8208962893594656, 113809776294348576, 1526808627197721792, 21533423236302943296

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388, A261317.

Column k=6 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 +x^3/3 +x^6/6) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 14 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 3, 6])))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

a[n_] := a[n] = If[n<0, 0, If[n == 0, 1, Sum[Product[n-i, {i, 1, j-1}]*a[n-j], {j, {1, 2, 3, 6}}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^3/3 +x^6/6], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 14 2019 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x+x^2/2+x^3/3+x^6/6) )) \\ G. C. Greubel, May 14 2019

m = 30; T = taylor(exp(x +x^2/2 +x^3/3 +x^6/6), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

E.g.f.: exp(x +x^2/2 +x^3/3 +x^6/6).

D-finite with recurrence a(n) -a(n-1) +(-n+1)*a(n-2) -(n-1)*(n-2)*a(n-3) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)=0. - R. J. Mathar, Jul 04 2023

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

Original entry on oeis.org

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

A261430 Number A(n,k) of permutations p of [n] without fixed points such that p^k = Id; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 9, 0, 15, 0, 0, 1, 0, 0, 2, 0, 0, 40, 0, 0, 0, 1, 0, 1, 0, 3, 24, 105, 0, 105, 0, 0, 1, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 9, 0, 175, 0, 2625, 2240, 945, 0, 0

Offset: 0

Alois P. Heinz, Aug 18 2015

			Square array A(n,k) begins:
  1, 1,   1,  1,    1,  1,    1,   1,    1, ...
  0, 0,   0,  0,    0,  0,    0,   0,    0, ...
  0, 0,   1,  0,    1,  0,    1,   0,    1, ...
  0, 0,   0,  2,    0,  0,    2,   0,    0, ...
  0, 0,   3,  0,    9,  0,    3,   0,    9, ...
  0, 0,   0,  0,    0, 24,   20,   0,    0, ...
  0, 0,  15, 40,  105,  0,  175,   0,  105, ...
  0, 0,   0,  0,    0,  0,  210, 720,    0, ...
  0, 0, 105,  0, 2625,  0, 4585,   0, 7665, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0+1,2-10 give: A000007, A001147, A052502, A052503, A052504, A261317, A261427, A261428, A261429, A261381.
Main diagonal gives A261431.
Cf. A008307.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,
      add(mul(n-i, i=1..j-1)*A(n-j, k), j=divisors(k) minus {1})))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[0, 0] = A[0, 1] = 1; A[, 0|1] = 0; A[n, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1, Sum[Product[n - i, {i, 1, j - 1}]*A[n - j, k], {j, Rest @ Divisors[k]}]]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 21 2017, after Alois P. Heinz *)

E.g.f. of column k: exp(Sum_{d|k, d>1} x^d/d).

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

A261381 Number of permutations sigma of [n] without fixed points such that sigma^10 = Id.

Original entry on oeis.org

1, 0, 1, 0, 3, 24, 15, 504, 105, 9072, 436401, 166320, 28750491, 3243240, 1307809503, 27965161224, 52309001745, 3795543015264, 2000776242465, 324424646818272, 17268536366932851, 22708075360010040, 3974396337125445231, 1436250980764880280, 548178165969608527353

Offset: 0

Alois P. Heinz, Aug 17 2015

nonn

			a(4) = 3: 2143, 3412, 4321:
a(5) = 24: 23451, 23514, 24153, 24531, 25134, 25413, 31452, 31524, 34251, 34512, 35214, 35421, 41253, 41532, 43152, 43521, 45123, 45231, 51234, 51423, 53124, 53412, 54132, 54213.
a(6) = 15: 214365, 215634, 216543, 341265, 351624, 361542, 432165, 456123, 465132, 532614, 546213, 564312, 632541, 645231, 654321.

Alois P. Heinz, Table of n, a(n) for n = 0..490

Column k=10 of A261430.

Cf. A001147, A052502, A052503, A052504, A053496, A261317.

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[2, 5, 10])))
    end:
seq(a(n), n=0..30);

E.g.f.: exp(x^2/2+x^5/5+x^10/10).

A052503 Number of permutations sigma of [2n] without fixed points such that sigma^4 = Id.

Original entry on oeis.org

1, 1, 9, 105, 2625, 76545, 3440745, 176080905, 12034447425, 922995698625, 87505195602825, 9203114782686825, 1141501848477415425, 155540530213013570625, 24232951756530007115625, 4112826185329479728735625, 781060320618828163499210625

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..280
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 28

Cf. A001472, A261317, A261381.

Bisection of column k=4 of A261430.

m:=40; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x^2/2 + x^4/4) )); [Factorial(2*n-2)*b[2*n-1]: n in [1..Floor((m-2)/2)]]; // G. C. Greubel, May 14 2019

spec := [S,{S=Set(Union(Cycle(Z,card=2),Cycle(Z,card=4)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nmax = 40}, CoefficientList[Series[Exp[x^2*(2 + x^2)/4], {x, 0, nmax}], x]*(Range[0, nmax])!][[1 ;; -1 ;; 2]] (* G. C. Greubel, May 14 2019 *)

x='x+O('x^40); v=Vec(serlaplace( exp(x^2/2 + x^4/4) )); vector(#v\2, n, v[2*n-1]) \\ G. C. Greubel, May 14 2019

m = 40; T = taylor(exp(x^2/2 + x^4/4), x, 0, 2*m+2); [factorial(2*n)*T.coefficient(x, 2*n) for n in (0..m)] # G. C. Greubel, May 14 2019

a(n) = (2n)! * [x^(2n)] exp(x^2/2 + x^4/4).
D-finite with recurrence a(n) +(-2*n+1)*a(n-1) -2*(n-1)*(2*n-1)*(2*n-3)*a(n-2)=0, with a(0)=1, a(1)=1, a(2)=9. - Corrected by R. J. Mathar, Feb 20 2020 to skip zeros.
a(n) = 2^n*Gamma(n+1/2)*A047974(n)/Pi^(1/2). - Mark van Hoeij, Oct 30 2011

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A053496 Number of degree-n permutations of order dividing 6.

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

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A261430 Number A(n,k) of permutations p of [n] without fixed points such that p^k = Id; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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

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A261381 Number of permutations sigma of [n] without fixed points such that sigma^10 = Id.

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A052503 Number of permutations sigma of [2n] without fixed points such that sigma^4 = Id.

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