OFFSET
0,4
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).
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
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 *)
PROG
(Magma) [n le 2 select 0 else n*(2^(n-1)-2): n in [0..40]]; // Vincenzo Librandi, Nov 18 2014
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001
STATUS
approved