A261430 -id:A261430 - 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

A261317 Number of permutations sigma of [n] without fixed points such that sigma^6 = Id.

Original entry on oeis.org

1, 0, 1, 2, 3, 20, 175, 210, 4585, 24920, 101745, 1266650, 13562395, 48588540, 1082015935, 9135376250, 63098660625, 1069777108400, 13628391601825, 88520971388850, 2134604966569075, 23945393042070500, 236084869688242575, 4893567386193135650, 72576130763294383225

Offset: 0

Alois P. Heinz, Aug 14 2015

nonn

			a(4) = 3: 2143, 3412, 4321.
a(5) = 20: 21453, 21534, 23154, 24513, 25431, 31254, 34152, 34521, 35124, 35412, 41523, 43251, 43512, 45132, 45213, 51432, 53214, 53421, 54123, 54231.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=6 of A261430.

Cf. A001147, A052502, A052503, A052504, A053496, A261381.

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, 3, 6])))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n < 0, 0, If[n == 0, 1, Sum[Product[n - i, {i, 1, j - 1}]*a[n - j], {j, {2, 3, 6}}]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 10 2018, from Maple *)

E.g.f.: exp(x^2*(x^4+2*x+3)/6).

D-finite with recurrence a(n) +(-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

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

A261431 Number of permutations p of [n] without fixed points such that p^n = Id.

Original entry on oeis.org

1, 0, 1, 2, 9, 24, 175, 720, 7665, 42560, 436401, 3628800, 70215145, 479001600, 7116730335, 88966701824, 1653438211425, 20922789888000, 457688776369825, 6402373705728000, 145083396337080201, 2457732174030848000, 55735573291977790575, 1124000727777607680000

Offset: 0

Alois P. Heinz, Aug 18 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Main diagonal of A261430.

Cf. A074759.

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:
a:= n-> A(n$2):
seq(a(n), n=0..25);

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, Divisors[k] ~Complement~ {1}}]]];
a[n_] := A[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

a(n) = n! * [x^n] exp(Sum_{d|n, d>1} x^d/d).

A261427 Number of permutations p of [7n] without fixed points such that p^7 = Id.

Original entry on oeis.org

1, 720, 889574400, 24825530695680000, 5290995845684330496000000, 5123434663327851951402516480000000, 16586604100059403377645257954741452800000000, 146550752102252281868362306608987740351496192000000000

Offset: 0

Alois P. Heinz, Aug 18 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..73

7-section of column k=7 of A261430.

Cf. A053497.

a:= proc(n) option remember; `if`(n=0, 1,
      mul(7*n-i, i=1..6)*a(n-1))
    end:
seq(a(n), n=0..10);

a(n) = (7n)! * [x^(7n)] exp(x^7/7).

A261428 Number of permutations p of [2n] without fixed points such that p^8 = Id.

Original entry on oeis.org

1, 1, 9, 105, 7665, 303345, 25893945, 1765268505, 345763843425, 42813526781025, 9399638261838825, 1573582072888650825, 563295733721953657425, 139523356060051359020625, 55722660999371761475705625, 17053184982967015188566885625, 9496879931794641573011009810625

Offset: 0

Alois P. Heinz, Aug 18 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Bisection of column k=8 of A261430.

Cf. A053498.

b:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*b(n-j), j=[2,4,8])))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..20);

a(n) = (2n)! * [x^(2n)] exp(x^2/2+x^4/4+x^8/8).

A261429 Number of permutations p of [3n] without fixed points such that p^9 = Id.

Original entry on oeis.org

1, 2, 40, 42560, 17987200, 8116908800, 43924225945600, 108050180446208000, 215140299047145472000, 2906668948375666073600000, 21059302309493030917734400000, 112131367456110324265700556800000, 2891761281909068919518711775232000000

Offset: 0

Alois P. Heinz, Aug 18 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..165

Trisection of column k=9 of A261430.

Cf. A053499.

b:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*b(n-j), j=[3,9])))
    end:
a:= n-> b(3*n):
seq(a(n), n=0..15);

a(n) = (3n)! * [x^(3n)] exp(x^3/3+x^9/9).

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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A261317 Number of permutations sigma of [n] without fixed points such that sigma^6 = 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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A261431 Number of permutations p of [n] without fixed points such that p^n = Id.

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A261427 Number of permutations p of [7n] without fixed points such that p^7 = Id.

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