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3^(n-3)*n*(n-1)*(n-2).
2

%I #33 Sep 02 2022 17:54:01

%S 0,0,0,6,72,540,3240,17010,81648,367416,1574640,6495390,25981560,

%T 101328084,386889048,1450833930,5356925280,19514513520,70252248672,

%U 250273635894,883318714920,3091615502220,10739295955080,37050571045026,127030529297232,433058622604200,1468633589701200

%N 3^(n-3)*n*(n-1)*(n-2).

%C The number of surjective functions f:{1,2,...,n}->{1,2,3} with a designated pre-image of 1, 2, and 3.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=748">Encyclopedia of Combinatorial Structures 748</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (12,-54,108,-81).

%F E.g.f.: x^3*exp(x)^3

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

%F a(n) = n!*sum(i+j+k=n, ijk/(i!j!k!)) - _Benoit Cloitre_, Nov 11 2004

%F G.f.: 6*x^3 / (3*x-1)^4. - _Colin Barker_, Jun 04 2013

%p spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t Range[0, 20]! CoefficientList[Series[(x Exp[x])^3, {x, 0, 20}], x]

%t LinearRecurrence[{12,-54,108,-81},{0,0,0,6},30] (* _Harvey P. Dale_, Sep 02 2022 *)

%o (PARI) a(n)=3^(n-3)*n*(n-1)*(n-2); /* _Joerg Arndt_, Sep 16 2012 */

%Y Cf. A001815.

%K easy,nonn

%O 0,4

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

%E Edited by _N. J. A. Sloane_, Dec 24 2010