A346085 -id:A346085 - cp's OEIS frontend

A110468 a(n) = (2*n + 1)!/(n + 1).

1, 3, 40, 1260, 72576, 6652800, 889574400, 163459296000, 39520825344000, 12164510040883200, 4644631106519040000, 2154334728240414720000, 1193170003333152768000000, 777776389315596582912000000, 589450799582646796969574400000, 513927415886120176107847680000000

Offset: 0

Paul Barry, Jul 21 2005

Convolution of (-1)^n*n! and n! with interpolated zeros suppressed.
Denominator of absolute value of coefficient of 1/(x+n^2) in the partial fraction decomposition of 1/(x+1)*1/(x+4)*..*1/(x+n^2). - Joris Roos (jorisr(AT)gmx.de), Aug 07 2009
With offset = 1: a(n) is the number of permutations of {1,2,...,2n} composed of two cycles of length n. - Geoffrey Critzer, Nov 11 2012

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A094310, A143623, A145877, A202768, A346085.

Table[(2n)!/(2n^2),{n,1,20}] (* Geoffrey Critzer, Nov 11 2012 *)

for(n=0,50, print1((2*n+1)!/(n+1), ", ")) \\ G. C. Greubel, Aug 28 2017

E.g.f.: log((1-x)*(1+x))/(-x).
a(n) = (2*n)!*Sum_{k = 0..2*n} (-1)^k/binomial(2*n, k).
a(n) = Sum_{k = 0..2*n} k!*(-1)^k*(2*n-k)!.
Sum_{n>=0} 1/a(n) = e/2. - Franz Vrabec, Jan 17 2008
(n+1)*a(n) + 2*(-n^2)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 15 2012
a(n) = Product_{i=1..n} (n+1-i)*(n+1+i). - Vaclav Kotesovec, Oct 21 2014
a(n) = A145877(2*n+2, n+1). - Alois P. Heinz, Apr 21 2017
a(n) = A346085(2*n+2, n+1). - Alois P. Heinz, Jul 04 2021
Sum_{n>=0} (-1)^n/a(n) = (cos(1) + sin(1))/2 = (1/2) * A143623. - Amiram Eldar, Feb 08 2022
a(p-1) == 1 (mod p), p a prime. - Peter Bala, Jul 29 2024
Sum_{n>=0} x^(2*n+1)/a(n) = (sinh(x) + x*cosh(x))/2. - Michael Somos, Jul 23 2025

Simpler definition from Robert Israel, Jul 20 2006

A346066 Sum of GCD of cycle lengths over all permutations of [n].

Original entry on oeis.org

0, 1, 3, 10, 45, 216, 1505, 9360, 84105, 730240, 7715169, 76204800, 1090114025, 11975040000, 185501455425, 2791872219136, 45361870178625, 690452066304000, 14415096609538625, 236887827111936000, 5448878874163974249, 108418310412206080000, 2381309423564793710625

Offset: 0

Alois P. Heinz, Jul 03 2021

nonn

			a(3) = 10 = 3+3+1+1+1+1: (123), (132), (1)(23), (13)(2), (12)(3), (1)(2)(3).

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

Cf. A060014 (the same for LCM), A346085.

b:= proc(n, g) option remember; `if`(n=0, g, add((j-1)!
      *b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);

b[n_, g_] := b[n, g] = If[n == 0, g, Sum[(j - 1)!*
     b[n - j, GCD[g, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A346085(n,k).

A346086 Number of permutations of [2n] such that the GCD of the cycle lengths equals 2.

Original entry on oeis.org

0, 1, 3, 105, 4725, 530145, 45270225, 12034447425, 2116670180625, 737902583042625, 219604524727012425, 137952599116097390625, 49583382753435146240625, 46991310794950147391390625, 25508895927267586991297765625, 24661803286201363305440202410625

Offset: 0

Alois P. Heinz, Jul 04 2021

nonn

			a(1) = 1: (12).
a(2) = 3: (12)(34), (13)(24), (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..225
Wikipedia, Permutation

Bisection of column k=2 of A346085 (even part).

b:= proc(n, g) option remember; `if`(n=0, `if`(g=2, 1, 0), `if`(g=1, 0,
       add(b(n-j, igcd(g, j))*binomial(n-1, j-1)*(j-1)!, j=2..n)))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..19);

b[n_, g_] := b[n, g] = If[n == 0, If[g == 2, 1, 0], If[g == 1, 0,
     Sum[b[n - j, GCD[g, j]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 2, n}]]];
a[n_] := b[2*n, 0];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)

a(n) = A346085(2n,2).

A359951 Number of permutations of [n] such that the GCD of the cycle lengths is a prime.

Original entry on oeis.org

0, 0, 1, 2, 3, 24, 145, 720, 4725, 22400, 602721, 3628800, 67692625, 479001600, 12924021825, 103953833984, 2116670180625, 20922789888000, 959231402754625, 6402373705728000, 257071215652932681, 3242340687872000000, 142597230222616430625, 1124000727777607680000

Offset: 0

Alois P. Heinz, Jan 19 2023

nonn

			a(2) = 1: (12).
a(3) = 2: (123), (132).
a(4) = 3: (12)(34), (13)(24), (14)(23).
a(5) = 24: (12345), (12354), (12435), (12453), (12534), (12543), (13245), (13254), (13425), (13452), (13524), (13542), (14235), (14253), (14325), (14352), (14523), (14532), (15234), (15243), (15324), (15342), (15423), (15432).

Alois P. Heinz, Table of n, a(n) for n = 0..451
Wikipedia, Permutation

Cf. A000040, A000142, A005225, A079128, A214003, A346085, A346086.

b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
      add(b(n-j, igcd(j, g))*(n-1)!/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);

b[n_, g_] := b[n, g] = If[n == 0, If[PrimeQ[g], 1, 0], Sum[b[n - j, GCD[j, g]]*(n - 1)!/(n - j)!, {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz *)

a(n) = Sum_{prime p <= n} A346085(n,p).

a(p) = (p-1)! for prime p.

cp's OEIS Frontend

A110468 a(n) = (2*n + 1)!/(n + 1).

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A346066 Sum of GCD of cycle lengths over all permutations of [n].

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A346086 Number of permutations of [2n] such that the GCD of the cycle lengths equals 2.

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Author

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Programs

Maple

Mathematica

Formula

A359951 Number of permutations of [n] such that the GCD of the cycle lengths is a prime.

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Author

Keywords

Examples

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Programs

Maple

Mathematica

Formula