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A132893 Triangle read by rows: T(n,k) is the number of paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., left factors of Motzkin paths) and having k peaks (i.e., UDs), 0 <= k <= floor(n/2).

%I A132893 #18 Oct 06 2019 08:51:04
%S A132893 1,2,4,1,9,4,21,13,1,50,40,6,121,118,27,1,296,340,106,8,730,965,381,
%T A132893 46,1,1812,2708,1296,220,10,4521,7535,4241,935,70,1,11328,20828,13482,
%U A132893 3676,395,12,28485,57266,41916,13658,1940,99,1
%N A132893 Triangle read by rows: T(n,k) is the number of paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., left factors of Motzkin paths) and having k peaks (i.e., UDs), 0 <= k <= floor(n/2).
%C A132893 Row n has 1 + floor(n/2) terms.
%C A132893 Row sums yield A005773.
%H A132893 Alois P. Heinz, <a href="/A132893/b132893.txt">Rows n = 0..200, flattened</a>
%H A132893 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.
%F A132893 T(n,0) = A091964(n).
%F A132893 Sum_{k=0..floor(n/2)} k*T(n,k) = A132894(n-1).
%F A132893 G.f.: G = G(t,z) satisfies z(1 - 3z + z^2 - tz^2)G^2 + (1 - 3z + z^2 - tz^2)G - 1 = 0 (see the Maple program for the explicit expression of G).
%e A132893 T(3,1)=4 because we have HUD, UDH, UDU and UUD.
%e A132893 Triangle starts:
%e A132893     1;
%e A132893     2;
%e A132893     4,   1;
%e A132893     9,   4;
%e A132893    21,  13,   1;
%e A132893    50,  40,   6;
%e A132893   121, 118,  27,   1;
%p A132893 G:=((-1+3*z-z^2+t*z^2+sqrt((1+z+z^2-t*z^2)*(1-3*z+z^2-t*z^2)))*1/2)/(z*(1-3*z+z^2-t*z^2)): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 12 do seq(coeff(P[n],t,j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
%p A132893 # second Maple program:
%p A132893 b:= proc(x, y, t) option remember; expand(`if`(y<0, 0,
%p A132893      `if`(x=0, 1, b(x-1, y, 1)+b(x-1, y-1, 1)*t+b(x-1, y+1, z))))
%p A132893     end:
%p A132893 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, 1)):
%p A132893 seq(T(n), n=0..15);  # _Alois P. Heinz_, Feb 01 2019
%t A132893 b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0, 0, If[x == 0, 1, b[x - 1, y, 1] + b[x - 1, y - 1, 1]*t + b[x - 1, y + 1, z]]]];
%t A132893 T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]] @ b[n, 0, 1];
%t A132893 T /@ Range[0, 15] // Flatten (* _Jean-François Alcover_, Oct 06 2019, after _Alois P. Heinz_ *)
%Y A132893 Cf. A005773, A091964, A132894.
%K A132893 nonn,tabf
%O A132893 0,2
%A A132893 _Emeric Deutsch_, Oct 08 2007