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A180119 a(n) = (n+2)! * Sum_{k = 1..n} 1/((k+1)*(k+2)). 7
0, 1, 6, 36, 240, 1800, 15120, 141120, 1451520, 16329600, 199584000, 2634508800, 37362124800, 566658892800, 9153720576000, 156920924160000, 2845499424768000, 54420176498688000, 1094805903679488000, 23112569077678080000, 510909421717094400000, 11802007641664880640000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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, 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)

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/(x-1)^3. (End)

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

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 *)

PROG

(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

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

CROSSREFS

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

Sequence in context: A066053 A153824 A001286 * A181964 A199422 A049431

Adjacent sequences:  A180116 A180117 A180118 * A180120 A180121 A180122

KEYWORD

nonn,easy,changed

AUTHOR

Gary Detlefs, Aug 10 2010

STATUS

approved

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Last modified April 20 20:11 EDT 2018. Contains 302827 sequences. (Running on oeis4.)