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
Number of surjections from {1,2,..,n+1} onto {1,2,...,n} (cf. A037960, A037961, A037962,...). - Roland Miyamoto, Apr 19 2026
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
H. W. Gould, ed. Jocelyn Quaintance, Combinatorial Identities, May 2010 (section 10, p.45).
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
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)
E.g.f.: x/(1-x)^3. (End)
a(n) = A001286(n+1) for n > 0. - M. F. Hasler, Apr 10 2018
From Amiram Eldar, Jan 08 2026: (Start)
Sum_{n>=1} 1/a(n) = 4 - 2*e - 2*gamma + 2*Ei(1), where e = A001113, gamma = A001620, and ExpIntegralEi(1) = A091725.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*gamma - 2/e - 2*Ei(-1), where Ei(-1) = -A099285. (End)
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
KEYWORD
nonn,easy
AUTHOR
Gary Detlefs, Aug 10 2010
STATUS
approved