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A114583 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k UHD's, where U=(1,1),H=(1,0),D=(1,-1) (0<=k<=floor(n/3)).

%I A114583 #24 Feb 01 2019 09:42:32
%S A114583 1,1,2,3,1,7,2,15,6,36,14,1,85,39,3,209,102,12,517,280,37,1,1303,758,
%T A114583 123,4,3312,2085,381,20,8510,5730,1194,76,1,22029,15849,3657,295,5,
%U A114583 57447,43914,11187,1056,30,150709,122090,33903,3734,135,1,397569,340104
%N A114583 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and having k UHD's, where U=(1,1),H=(1,0),D=(1,-1) (0<=k<=floor(n/3)).
%C A114583 Row n contains 1+floor(n/3) terms. Row sums are the Motzkin numbers (A001006). Column 1 yields A114584. Sum(k*T(n,k),k=0..floor(n/3))=A005717(n-2).
%H A114583 Alois P. Heinz, <a href="/A114583/b114583.txt">Rows n = 0..300, flattened</a>
%H A114583 Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Barnabei/barnabei5.html">Motzkin and Catalan Tunnel Polynomials</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
%H A114583 Zhuang, Yan. <a href="https://arxiv.org/abs/1508.02793">A generalized Goulden-Jackson cluster method and lattice path enumeration</a>, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2.
%F A114583 G.f.=G=G(t, z) satisfies G=1+zG+z^2*G(tz-z+G).
%e A114583 T(5,1)=6 because we have HH(UHD), UD(UHD), (UHD)HH, (UHD)UD, H(UHD)H and U(UHD)D, where U=(1,1),H=(1,0),D=(1,-1) (the UHD's are shown between parentheses).
%e A114583 Triangle begins:
%e A114583    1;
%e A114583    1;
%e A114583    2;
%e A114583    3,  1;
%e A114583    7,  2;
%e A114583   15,  6;
%e A114583   36, 14, 1;
%e A114583   ...
%p A114583 G:=(1-z-t*z^3+z^3-sqrt((1-3*z+z^3-t*z^3)*(1+z+z^3-t*z^3)))/2/z^2: Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 17 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 17 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/3)) od; # yields sequence in triangular form
%p A114583 # second Maple program:
%p A114583 b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
%p A114583      `if`(x=0, 1, b(x-1, y, `if`(t=1, 2, 0))+b(x-1, y-1, 0)*
%p A114583      `if`(t=2, z, 1)+b(x-1, y+1, 1))))
%p A114583     end:
%p A114583 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
%p A114583 seq(T(n), n=0..15);  # _Alois P. Heinz_, Feb 01 2019
%t A114583 CoefficientList[#, t]& /@ CoefficientList[(1 - z - t z^3 + z^3 - Sqrt[(1 - 3z + z^3 - t z^3)(1 + z + z^3 - t z^3)])/2/z^2 + O[z]^17, z] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)
%Y A114583 Cf. A001006, A114584, A115717.
%K A114583 nonn,tabf
%O A114583 0,3
%A A114583 _Emeric Deutsch_, Dec 09 2005