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 1-4 of 4 results.

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

Original entry on oeis.org

1, 3, 40, 1260, 72576, 6652800, 889574400, 163459296000, 39520825344000, 12164510040883200, 4644631106519040000, 2154334728240414720000, 1193170003333152768000000, 777776389315596582912000000, 589450799582646796969574400000, 513927415886120176107847680000000
Offset: 0

Views

Author

Paul Barry, Jul 21 2005

Keywords

Comments

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

Crossrefs

Programs

  • Mathematica
    Table[(2n)!/(2n^2),{n,1,20}] (* Geoffrey Critzer, Nov 11 2012 *)
  • PARI
    for(n=0,50, print1((2*n+1)!/(n+1), ", ")) \\ G. C. Greubel, Aug 28 2017

Formula

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

Extensions

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

Views

Author

Alois P. Heinz, Jul 03 2021

Keywords

Examples

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

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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

Views

Author

Alois P. Heinz, Jul 04 2021

Keywords

Examples

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

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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

Views

Author

Alois P. Heinz, Jan 19 2023

Keywords

Examples

			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).
		

Crossrefs

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

a(n) = Sum_{prime p <= n} A346085(n,p).
a(p) = (p-1)! for prime p.
Showing 1-4 of 4 results.