A180119 a(n) = (n+2)! * Sum_{k = 1..n} 1/((k+1)*(k+2)).

0, 1, 6, 36, 240, 1800, 15120, 141120, 1451520, 16329600, 199584000, 2634508800, 37362124800, 566658892800, 9153720576000, 156920924160000, 2845499424768000, 54420176498688000, 1094805903679488000, 23112569077678080000, 510909421717094400000, 11802007641664880640000

Offset: 0

Author

Gary Detlefs, Aug 10 2010

Keywords

• nonn
• easy

Comments

In general, a sequence of the form (n+x+2)! * Sum_{k = 1..n} (k+x)!/(k+x+2)! will have a closed form of (n+x+2)!*n/((x+2)*(n+2+x)).

0 followed by A001286. - R. J. Mathar, Aug 13 2010

Sum of the entries in all cycles of all permutations of [n]. - Alois P. Heinz, Apr 19 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 209. Book's website
• H. W. Gould, ed. J. Quaintance, Combinatorial Identities, May 2010 (section 10, p.45).

Crossrefs

Cf. A000142, A000217, A001286, A037960, A037961, A285439, A285793.

Programs

Magma

[n*Factorial(n+2)/(2*(n+2)): n in [0..25]]; // Vincenzo Librandi, Feb 20 2017

Maple

a:= n-> n*(n+2)!/(2*(n+2)): seq(a(n), n=0..20);

Mathematica

Table[n (n + 2)! / (2 (n + 2)), {n, 0, 30}] (* Vincenzo Librandi, Feb 20 2017 *)

PARI

a(n) = (n+2)! * sum(k=1, n,1/((k+1)*(k+2))); \\ Michel Marcus, Jan 10 2015

PARI

apply( A180119(n)=(n+1)!\2*n, [0..20]) \\ M. F. Hasler, Apr 10 2018

Formula

a(n) = n*(n+1)!/2. [Simplified by M. F. Hasler, Apr 10 2018]

a(n) = (n+1)! * Sum_{k = 2..n} (1/(k^2+k)), with offset 1. - Gary Detlefs, Sep 15 2011

a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*k^(n+1) = (1/(2*x + 1))*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(x*n + k)^(n+1), for arbitrary x != -1/2. - Peter Bala, Feb 19 2017

From Alois P. Heinz, Apr 19 2017: (Start)

a(n) = A000142(n) * A000217(n) = Sum_{k=1..n} A285439(n,k).

E.g.f.: x/(1-x)^3. (End)

a(n) = A001286(n+1) for n > 0. - M. F. Hasler, Apr 10 2018