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A108443 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k triple descents (i.e., ddd's).

%I A108443 #17 Jul 26 2017 03:16:31
%S A108443 1,2,6,3,1,21,24,15,5,1,80,150,145,84,31,7,1,322,857,1145,949,528,202,
%T A108443 53,9,1,1347,4692,8096,8801,6598,3551,1394,398,81,11,1,5798,25102,
%U A108443 53457,72338,68594,47805,25092,10019,3040,692,115,13,1,25512,132484,337132
%N A108443 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k triple descents (i.e., ddd's).
%C A108443 Row n has 2n-1 terms (n >= 1). Row sums yield A027307. Column 0 yields A106228.
%H A108443 Alois P. Heinz, <a href="/A108443/b108443.txt">Rows n = 0..100, flattened</a>
%H A108443 Emeric Deutsch, <a href="http://www.jstor.org/stable/2589192">Problem 10658: Another Type of Lattice Path</a>, American Math. Monthly, 107, 2000, 368-370.
%F A108443 G.f. G = G(t,z) satisfies G = 1 + z(t + z - tz)^2*G^3 + z(2-t)(t + z - tz)G^2 + 2z(1-t)G.
%e A108443 T(2,1) = 3 because we have uUddd, Uuddd and UdUddd.
%e A108443 Triangle begins:
%e A108443     1;
%e A108443     2;
%e A108443     6,    3,    1;
%e A108443    21,   24,   15,    5,    1;
%e A108443    80,  150,  145,   84,   31,    7,    1;
%e A108443   322,  857, 1145,  949,  528,  202,   53,    9,    1;
%p A108443 b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
%p A108443       `if`(x=0, 1, b(x-1, y-1, min(2, t+1))*`if`(t=2, z, 1)+
%p A108443        b(x-1, y+2, 0)+b(x-2, y+1, 0))))
%p A108443     end:
%p A108443 T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(3*n, 0$2)):
%p A108443 seq(T(n), n=0..8);  # _Alois P. Heinz_, Oct 06 2015
%t A108443 b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0 || y > x, 0, If[x == 0, 1, b[x - 1, y - 1, Min[2, t + 1]]*If[t == 2, z, 1] + b[x - 1, y + 2, 0] + b[x - 2, y + 1, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[3*n, 0, 0]]; Table[T[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Dec 21 2016, after _Alois P. Heinz_ *)
%o A108443 (PARI) {T(n,k)=local(G=1+z*O(z^n)+t*O(t^k));for(k=1,n, G=1+z*(t+z-t*z)^2*G^3+z*(2-t)*(t+z-t*z)*G^2+2*z*(1-t)*G); polcoeff(polcoeff(G,n,z),k,t)}
%Y A108443 Cf. A027307, A106228.
%K A108443 nonn,tabf
%O A108443 0,2
%A A108443 _Emeric Deutsch_ and _Paul D. Hanna_, Jun 10 2005