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A289693 The number of partitions of [n] with exactly 3 blocks without peaks.

%I A289693 #13 Jul 18 2023 10:38:09
%S A289693 0,0,1,3,9,27,75,197,503,1263,3132,7695,18784,45649,110585,267276,
%T A289693 644907,1554208,3742321,9005265,21659603,52078400,125186565,300870586,
%U A289693 723010749,1737273406,4174084259,10028409724,24092769583,57880137331
%N A289693 The number of partitions of [n] with exactly 3 blocks without peaks.
%H A289693 T. Mansour and M. Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Shattuck/shattuck3.html">Counting Peaks and Valleys in a Partition of a Set</a>, J. Int. Seq. 13 (2010), 10.6.8, Table 1.
%H A289693 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,24,-29,25,-17,9,-3,1).
%F A289693 From _Colin Barker_, Nov 07 2017: (Start)
%F A289693 G.f.: x^3*(1 - x + x^2)*(1 - 2*x + 3*x^2 - x^3 + x^4) / ((1 - x)*(1 - 2*x + x^2 - x^3)*(1 - 3*x + 3*x^2 - 4*x^3 + x^4 - x^5)).
%F A289693 a(n) = 6*a(n-1) - 15*a(n-2) + 24*a(n-3) - 29*a(n-4) + 25*a(n-5) - 17*a(n-6) + 9*a(n-7) - 3*a(n-8) + a(n-9) for n>9.
%F A289693 (End)
%p A289693 with(orthopoly) :
%p A289693 nmax := 10:
%p A289693 tpr := 1+x^2/2 :
%p A289693 k := 3:
%p A289693 g := x^k ;
%p A289693 for j from 1 to k do
%p A289693         if j> 1 then
%p A289693                 g := g*( U(j-1,tpr)-(1+x)*U(j-2,tpr)) / ((1-x)*U(j-1,tpr)-U(j-2,tpr)) ;
%p A289693         else
%p A289693                 # note that U(-1,.)=0, U(0,.)=1
%p A289693                 g := g* U(j-1,tpr) / ((1-x)*U(j-1,tpr)) ;
%p A289693         end if;
%p A289693 end do:
%p A289693 simplify(%) ;
%p A289693 taylor(g,x=0,nmax+1) ;
%p A289693 gfun[seriestolist](%) ; # _R. J. Mathar_, Mar 11 2021
%Y A289693 Cf. A289692, A289694.
%K A289693 nonn
%O A289693 1,4
%A A289693 _R. J. Mathar_, Jul 09 2017