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A108441 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 having k U=(1,2) steps among the steps leading to the first d step.

%I A108441 #12 Jan 29 2016 10:25:02
%S A108441 1,1,1,3,6,1,15,39,11,1,97,284,100,16,1,721,2249,888,186,21,1,5827,
%T A108441 18890,7977,1952,297,26,1,49759,165519,72991,19731,3601,433,31,1,
%U A108441 441729,1496696,680096,196864,40586,5960,594,36,1,4035937,13865297,6439656
%N A108441 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 having k U=(1,2) steps among the steps leading to the first d step.
%H A108441 Alois P. Heinz, <a href="/A108441/b108441.txt">Rows n = 0..140, flattened</a>
%H A108441 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 A108441 G.f.: G(t, z)=1/(1-zA-tzA^2)-1, where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
%e A108441 T(2,1)=6 because we have uUddd, Uddud, UddUdd, Ududd, UdUddd and Uuddd.
%e A108441 Triangle begins:
%e A108441 1;
%e A108441 1,1;
%e A108441 3,6,1;
%e A108441 15,39,11,1;
%e A108441 97,284,100,16,1;
%p A108441 A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=1/(1-z*A-t*z*A^2): Gser:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 9 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 9 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
%p A108441 # second Maple program:
%p A108441 b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
%p A108441       `if`(x=0, 1, b(x-1, y-1, false)+b(x-1, y+2, t)*
%p A108441       `if`(t, z, 1)+b(x-2, y+1, t))))
%p A108441     end:
%p A108441 T:= n-> (p-> seq(coeff(p, z, i), i=0..n))(b(3*n, 0, true)):
%p A108441 seq(T(n), n=0..10);  # _Alois P. Heinz_, Oct 06 2015
%t A108441 b[x_, y_, t_] := b[x, y, t] = Expand[If[y<0 || y>x, 0, If[x==0, 1, b[x-1, y - 1, False] + b[x-1, y+2, t]*If[t, z, 1] + b[x-2, y+1, t]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n}]][ b[3*n, 0, True]]; Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 29 2016, after _Alois P. Heinz_ *)
%Y A108441 Row sums yield A027307. Column 0 yields A108442.
%K A108441 nonn,tabl
%O A108441 0,4
%A A108441 _Emeric Deutsch_, Jun 08 2005