cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

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

Keywords

Comments

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.

References

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

Links

Crossrefs

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.

Programs

  • GAP
    List([0..20], n-> Factorial(3*n)/(3^n*Factorial(n))) # G. C. Greubel, May 14 2019
  • Magma
    [Factorial(3*n)/(3^n*Factorial(n)): n in [0..20]]; // G. C. Greubel, May 14 2019
  • Maple
    spec := [S,{S=Set(Union(Cycle(Z,card=3)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    Table[(3*n)!/(3^n*n!), {n, 0, 20}] (* G. C. Greubel, May 14 2019 *)
  • PARI
    {a(n) = (3*n)!/(3^n*n!)}; \\ G. C. Greubel, May 14 2019
  • Sage
    [factorial(3*n)/(3^n*factorial(n)) for n in (0..20)] # G. C. Greubel, May 14 2019

Formula

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)

Extensions

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

Views

Author

Alois P. Heinz, Aug 18 2015

Keywords

Examples

			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, ...

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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

Views

Author

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

Keywords

Comments

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.

Links

Crossrefs

Cf. A001147, A052502, A060706, A261317, A261381.
Quintisection of column k=5 of A261430.

Programs

  • GAP
    List([0..15], n-> Factorial(5*n)/(5^n*Factorial(n))) # G. C. Greubel, May 14 2019
  • Magma
    [Factorial(5*n)/(5^n*Factorial(n)): n in [0..15]]; // G. C. Greubel, May 14 2019
  • Maple
    spec := [S,{S=Set(Union(Cycle(Z,card=5)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    nn = 50; Select[Range[0, nn]! CoefficientList[Series[Exp[x^5/5], {x, 0, nn}], x], # > 0 &]  (* Geoffrey Critzer, Aug 19 2012 *)
  • PARI
    {a(n) = (5*n)!/(5^n*n!)}; \\ G. C. Greubel, May 14 2019
  • Sage
    [factorial(5*n)/(5^n*factorial(n)) for n in (0..15)] # G. C. Greubel, May 14 2019

Formula

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

Views

Author

Alois P. Heinz, Aug 14 2015

Keywords

Examples

			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.

Links

Crossrefs

Column k=6 of A261430.
Cf. A001147, A052502, A052503, A052504, A053496, A261381.

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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

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

Views

Author

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

Keywords

Links

Crossrefs

Cf. A001472, A261317, A261381.
Bisection of column k=4 of A261430.

Programs

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

Formula

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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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