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.
%I A180119 #68 Dec 07 2024 02:01:12
%S A180119 0,1,6,36,240,1800,15120,141120,1451520,16329600,199584000,2634508800,
%T A180119 37362124800,566658892800,9153720576000,156920924160000,
%U A180119 2845499424768000,54420176498688000,1094805903679488000,23112569077678080000,510909421717094400000,11802007641664880640000
%N A180119 a(n) = (n+2)! * Sum_{k = 1..n} 1/((k+1)*(k+2)).
%C A180119 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)).
%C A180119 0 followed by A001286. - _R. J. Mathar_, Aug 13 2010
%C A180119 Sum of the entries in all cycles of all permutations of [n]. - _Alois P. Heinz_, Apr 19 2017
%H A180119 Vincenzo Librandi, <a href="/A180119/b180119.txt">Table of n, a(n) for n = 0..300</a>
%H A180119 A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 209. <a href="http://tohbook.info">Book's website</a>
%H A180119 H. W. Gould, ed. J. Quaintance, <a href="http://www.math.wvu.edu/~hgould/Vol.4.PDF">Combinatorial Identities</a>, May 2010 (section 10, p.45).
%F A180119 a(n) = n*(n+1)!/2. [Simplified by _M. F. Hasler_, Apr 10 2018]
%F A180119 a(n) = (n+1)! * Sum_{k = 2..n} (1/(k^2+k)), with offset 1. - _Gary Detlefs_, Sep 15 2011
%F A180119 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
%F A180119 From _Alois P. Heinz_, Apr 19 2017: (Start)
%F A180119 a(n) = A000142(n) * A000217(n) = Sum_{k=1..n} A285439(n,k).
%F A180119 E.g.f.: x/(1-x)^3. (End)
%F A180119 a(n) = A001286(n+1) for n > 0. - _M. F. Hasler_, Apr 10 2018
%p A180119 a:= n-> n*(n+2)!/(2*(n+2)): seq(a(n), n=0..20);
%t A180119 Table[n (n + 2)! / (2 (n + 2)), {n, 0, 30}] (* _Vincenzo Librandi_, Feb 20 2017 *)
%o A180119 (PARI) a(n) = (n+2)! * sum(k=1, n,1/((k+1)*(k+2))); \\ _Michel Marcus_, Jan 10 2015
%o A180119 (PARI) apply( A180119(n)=(n+1)!\2*n, [0..20]) \\ _M. F. Hasler_, Apr 10 2018
%o A180119 (Magma) [n*Factorial(n+2)/(2*(n+2)): n in [0..25]]; // _Vincenzo Librandi_, Feb 20 2017
%Y A180119 Cf. A000142, A000217, A001286, A037960, A037961, A285439, A285793.
%K A180119 nonn,easy
%O A180119 0,3
%A A180119 _Gary Detlefs_, Aug 10 2010