A001804 a(n) = n! * C(n,2). (Formerly M2109 N0834)

2, 18, 144, 1200, 10800, 105840, 1128960, 13063680, 163296000, 2195424000, 31614105600, 485707622400, 7933224499200, 137305808640000, 2510734786560000, 48373490221056000, 979563176976384000, 20801312169910272000, 462251381553561600000

Offset

2,1

Comments

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

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

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

References

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.

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

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

Links

T. D. Noe, Table of n, a(n) for n = 2..100

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Index entries for sequences related to factorial numbers

Formula

E.g.f.: x^2/(1-x)^3. - Geoffrey Critzer, Aug 19 2012

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

From Ilya Gutkovskiy, Jan 20 2017: (Start)

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

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

Maple

seq(n!*binomial(n, 2), n=2..20); # Emeric Deutsch, Oct 01 2006
a:=n->sum((n-j)*n!, j=1..n): seq(a(n), n=2..22); # Zerinvary Lajos, Apr 29 2007
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

Mathematica

Table[n! Binomial[n, 2], {n, 2, 20}] (* T. D. Noe, Aug 10 2012 *)

Prog

(PARI) a(n) = n!*binomial(n, 2); \\ Michel Marcus, Mar 14 2016

Crossrefs

Cf. A001809, A120434.

Sequence in context: A193446 A226733 A220244 * A277182 A052640 A290215

Adjacent sequences: A001801 A001802 A001803 * A001805 A001806 A001807

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved