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A168583 The number of ways of partitioning the multiset {1,1,2,3,...,n-1} into exactly three nonempty parts.

%I A168583 #16 Sep 08 2022 08:45:49
%S A168583 1,4,16,58,196,634,1996,6178,18916,57514,174076,525298,1582036,
%T A168583 4758394,14299756,42948418,128943556,387027274,1161475036,3485211538,
%U A168583 10457207476,31374768154,94130595916,282404370658,847238277796,2541765165034,7625396158396
%N A168583 The number of ways of partitioning the multiset {1,1,2,3,...,n-1} into exactly three nonempty parts.
%C A168583 The number of ways of partitioning the multiset {1, 1, 2, 3, ..., n-1} into exactly two, four and five nonempty parts are given in A083329, A168584 and A168585, respectively.
%H A168583 M. Griffiths, I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths/griffiths11.html">A generalization of Stirling Numbers of the Second Kind via a special multiset</a>, JIS 13 (2010) #10.2.5.
%H A168583 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F A168583 For a>=3, a(n) = 3^(n-2) - 3*2^(n-3) + 1.
%F A168583 E.g.f.: 3*e^(3x) - 3*e^(2x) + e^x (shifted).
%F A168583 O.g.f.: x^3*(1-2x+3x^2)/((1-x)*(1-2x)*(1-3x)).
%F A168583 a(n) = A126644(n-3). - _R. J. Mathar_, Dec 11 2009
%e A168583 The partitions of {1,1,2,3} into exactly three nonempty parts are {{1},{1},{2,3}}, {{1},{2},{1,3}}, {{1},{3},{1,2}} and {{2},{3},{1,1}}.
%p A168583 A168583:=n->3^(n-2)-3*2^(n-3)+1: seq(A168583(n), n=3..40); # _Wesley Ivan Hurt_, Dec 12 2015
%t A168583 f1[n_] := 3^(n - 2) - 3 2^(n - 3) + 1; Table[f1[n], {n, 3, 25}]
%o A168583 (Magma) [3^(n-2) - 3*2^(n-3) + 1: n in [3..35]]; // _Vincenzo Librandi_, Dec 12 2015
%Y A168583 Cf. A083329, A126644, A168584, A168585.
%K A168583 nonn,easy
%O A168583 3,2
%A A168583 _Martin Griffiths_, Nov 30 2009