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

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

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Crossrefs

Cf. A001809, A120434.

Programs

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 *)

PARI

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

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)