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%I #45 Jun 12 2023 14:52:14
%S 0,0,0,6,24,70,180,434,1008,2286,5100,11242,24552,53222,114660,245730,
%T 524256,1114078,2359260,4980698,10485720,22020054,46137300,96468946,
%U 201326544,419430350,872415180,1811939274,3758096328,7784628166,16106127300,33285996482
%N a(n) = 2*n * Stirling2(n-1,2).
%C 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
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=705">Encyclopedia of Combinatorial Structures 705</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4).
%F E.g.f.: x*exp(x)^2 - 2*x*exp(x) + x.
%F 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)}.
%F a(n) = Sum_{k=3..n} n*2^(k-2). - _Zerinvary Lajos_, Oct 09 2006
%F a(n) = n*(2^(n-1)-2) = n*A000918(n-1), n >= 3. - _Mitch Harris_, Oct 25 2006
%F 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
%F a(n) = Sum_{j=1..n} ( Sum_{i=2..n-1} (j+1)*2^(j-i-1) ). - _Wesley Ivan Hurt_, Nov 17 2014
%F a(n) = n*(2^n-4)/2, n > 1. - _Wesley Ivan Hurt_, Nov 17 2014
%F a(n) = 2*A260006(n-2). - _R. J. Mathar_, Apr 26 2017
%p spec := [S,{B=Set(Z,1 <= card),S=Prod(Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p g := taylor(exp(x)^2*x-2*x*exp(x)+x,x=0,121): q := seq(coeff(g,x,i)*i!,i=0..120);
%t Table[If[n < 3, 0, (n*(2^n - 3) - n)/2], {n, 0, 200}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 30 2011 *)
%t LinearRecurrence[{6,-13,12,-4},{0,0,0,6,24,70},40] (* _Harvey P. Dale_, Aug 30 2017 *)
%o (Magma) [n le 2 select 0 else n*(2^(n-1)-2): n in [0..40]]; // _Vincenzo Librandi_, Nov 18 2014
%Y Cf. A008277, A000918, A260006.
%K easy,nonn
%O 0,4
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001
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