%I M2109 N0834 #44 Dec 19 2021 10:04:29

%S 2,18,144,1200,10800,105840,1128960,13063680,163296000,2195424000,

%T 31614105600,485707622400,7933224499200,137305808640000,

%U 2510734786560000,48373490221056000,979563176976384000,20801312169910272000,462251381553561600000

%N a(n) = n! * C(n,2).

%C Number of big descents in all permutations of [n+1]. A big descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) >= 2. Example: a(2)=2 because there are 2 big descents in the permutations 123, 132, 213, 23\1, 3\12, 321 of {1,2,3} (shown by a \). a(n)=Sum(k*A120434(n+1,k),k=0..n-1). - _Emeric Deutsch_, Oct 01 2006

%C a(n)/2 counts the total number of inversions in all the permutations of the set [n]; see A001809. - _Peter Bala_, Feb 28 2013

%C Equivalently, number of mappings f from a set X of n elements into itself such that f(X) has n-1 elements. - _Robert FERREOL_, Mar 14 2016

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001804/b001804.txt">Table of n, a(n) for n = 2..100</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F E.g.f.: x^2/(1-x)^3. - _Geoffrey Critzer_, Aug 19 2012

%F a(n) = 2 * A001809(n).

%F From _Ilya Gutkovskiy_, Jan 20 2017: (Start)

%F a(n) ~ sqrt(Pi/2)*n^(n+5/2)/exp(n).

%F Sum_{n>=2} 1/a(n) = 2*(3 - exp(1)) = 0.563436343081909529... (End)

%p seq(n!*binomial(n,2),n=2..20); # _Emeric Deutsch_, Oct 01 2006

%p a:=n->sum((n-j)*n!, j=1..n): seq(a(n), n=2..22); # _Zerinvary Lajos_, Apr 29 2007

%p restart: G(x):=x^2/(1-x)^3: f[0]:=G(x): for n from 1 to 18 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=2..16); # _Zerinvary Lajos_, Apr 01 2009

%t Table[n! Binomial[n, 2], {n, 2, 20}] (* _T. D. Noe_, Aug 10 2012 *)

%o (PARI) a(n) = n!*binomial(n, 2); \\ _Michel Marcus_, Mar 14 2016

%Y Cf. A001809, A120434.

%K nonn,easy

%O 2,1

%A _N. J. A. Sloane_