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A168605 Number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly three nonempty parts.

%I A168605 #16 Feb 07 2021 07:04:54
%S A168605 1,2,8,30,104,342,1088,3390,10424,31782,96368,291150,877544,2640822,
%T A168605 7938848,23849310,71613464,214971462,645176528,1936053870,5809210184,
%U A168605 17429727702,52293377408,156888520830,470682339704,1412080573542
%N A168605 Number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly three nonempty parts.
%C A168605 The number of ways of partitioning the multiset {1, 1, 1, 2, 3, ..., n-1} into exactly two and four nonempty parts are given in A168604 and A168606, respectively.
%H A168605 G. C. Greubel, <a href="/A168605/b168605.txt">Table of n, a(n) for n = 3..1000</a>
%H A168605 M. Griffiths and 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 A168605 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F A168605 a(n) = (5*3^(n-3) - 3*2^(n-2) + 3)/3 for n >= 4, with a(3) = 1.
%F A168605 The shifted e.g.f. is (5*exp(3*x) - 6*exp(2*x) + 3*exp(x) + 1)/3.
%F A168605 G.f.: x^3*(1 -4*x +7*x^2 -2*x^3)/((1-x)*(1-2*x)*(1-3*x)).
%t A168605 a[n_]:= If[n==3, 1, (5*3^(n-3) - 3*2^(n-2) + 3)/3]; Table[a[n], {n, 3, 30}]
%o A168605 (Sage) [1]+[(5*3^(n-3) -3*2^(n-2) +3)/3 for n in (4..30)] # _G. C. Greubel_, Feb 07 2021
%o A168605 (Magma) [1] cat [(5*3^(n-3) -3*2^(n-2) +3)/3: n in [4..30]]; // _G. C. Greubel_, Feb 07 2021
%Y A168605 Cf. A168604, A168606.
%K A168605 nonn,easy
%O A168605 3,2
%A A168605 _Martin Griffiths_, Dec 01 2009
%E A168605 Last element of the multiset in the definition corrected by _Martin Griffiths_, Dec 02 2009