A052749 a(n) = 2*n * Stirling2(n-1,2).

0, 0, 0, 6, 24, 70, 180, 434, 1008, 2286, 5100, 11242, 24552, 53222, 114660, 245730, 524256, 1114078, 2359260, 4980698, 10485720, 22020054, 46137300, 96468946, 201326544, 419430350, 872415180, 1811939274, 3758096328, 7784628166, 16106127300, 33285996482

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

a(n) is the number of ordered set partitions of an n-set into 3 nonempty sets such that the first set contains exactly one element. a(5) = 70 since the ordered set partitions are the following: 20 of type {1},{2,3,4},{5}; 30 of type {1},{2,3},{4,5}; 20 of type {1},{2},{3,4,5}. - Enrique Navarrete, Jun 11 2023

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 705
• Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Crossrefs

Cf. A008277, A000918, A260006.

Programs

Magma

[n le 2 select 0 else n*(2^(n-1)-2): n in [0..40]]; // Vincenzo Librandi, Nov 18 2014

Maple

spec := [S,{B=Set(Z,1 <= card),S=Prod(Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
g := taylor(exp(x)^2*x-2*x*exp(x)+x,x=0,121): q := seq(coeff(g,x,i)*i!,i=0..120);

Mathematica

Table[If[n < 3, 0, (n*(2^n - 3) - n)/2], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 30 2011 *)
LinearRecurrence[{6,-13,12,-4},{0,0,0,6,24,70},40] (* Harvey P. Dale, Aug 30 2017 *)

Formula

E.g.f.: x*exp(x)^2 - 2*x*exp(x) + x.

Recurrence: {a(1)=0, a(2)=0, a(3)=6, (2*n^2+6*n+4)*a(n)+(-6*n-3*n^2)*a(n+1)+(n^2+n)*a(n+2)}.

a(n) = Sum_{k=3..n} n*2^(k-2). - Zerinvary Lajos, Oct 09 2006

a(n) = n*(2^(n-1)-2) = n*A000918(n-1), n >= 3. - Mitch Harris, Oct 25 2006

O.g.f.: 2*x^3*(3-6*x+2*x^2)/((-1+x)^2*(-1+2*x)^2). - R. J. Mathar, Dec 05 2007

a(n) = Sum_{j=1..n} ( Sum_{i=2..n-1} (j+1)*2^(j-i-1) ). - Wesley Ivan Hurt, Nov 17 2014

a(n) = n*(2^n-4)/2, n > 1. - Wesley Ivan Hurt, Nov 17 2014

a(n) = 2*A260006(n-2). - R. J. Mathar, Apr 26 2017

Extensions

Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001