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A190170 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k UHD's starting at level 0; here U=(1,1), H=(1,0), and D=(1,-1).

%I A190170 #9 May 29 2022 03:21:21
%S A190170 1,1,1,1,1,2,2,5,3,12,4,1,27,7,3,60,16,6,135,39,10,1,309,92,18,4,717,
%T A190170 212,39,10,1680,488,94,20,1,3966,1135,228,39,5,9423,2670,543,84,15,
%U A190170 22518,6336,1282,200,35,1,54091,15132,3036,492,75,6,130540,36327,7245,1203,166,21
%N A190170 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k UHD's starting at level 0; here U=(1,1), H=(1,0), and D=(1,-1).
%C A190170 Row n contains 1+floor(n/3) entries.
%C A190170 Sum of entries in row n = A004148(n).
%C A190170 T(n,0)=A190171(n).
%C A190170 Sum(k*T(n,k),k>=0)=A089735(n-3).
%F A190170 G.f. G=G(t,z) is obtained by elimitaing S from the equations G=1+zG+z^2*G(S-1-z+tz) and S=1+zS+z^2*S(S-1).
%e A190170 T(6,2)=1 because we have UHDUHD.
%e A190170 Triangle starts:
%e A190170 1;
%e A190170 1;
%e A190170 1;
%e A190170 1,1;
%e A190170 2,2;
%e A190170 5,3;
%e A190170 12,4,1;
%e A190170 27,7,3;
%p A190170 p1 := G-1-z*G-z^2*G*(S-1-z+t*z): p2 := S-1-z*S-z^2*S*(S-1): r := resultant(p1, p2, S): g := RootOf(r, G): Gser := simplify(series(g, z = 0, 21)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
%Y A190170 Cf. A004148, A190171, A089735.
%K A190170 nonn,tabf
%O A190170 0,6
%A A190170 _Emeric Deutsch_, May 06 2011