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%I #13 Jan 29 2016 10:24:28
%S 1,1,1,7,2,1,47,15,3,1,361,108,24,4,1,2977,865,184,34,5,1,25775,7334,
%T 1533,276,45,6,1,231103,64767,13359,2387,385,57,7,1,2127409,589368,
%U 120376,21368,3450,512,70,8,1,19990241,5488033,1112424,196484,31706
%N 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 hills of the form ud (a hill is either a ud or a Udd starting at the x-axis).
%C Row sums yield A027307. T(n,0)=A108434(n). A027307, A108432, A108433, A108434.
%H Alois P. Heinz, <a href="/A108433/b108433.txt">Rows n = 0..140, flattened</a>
%H 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 G.f.: 1/(1-tz+z-zA-zA^2), where A=1+zA^2+zA^3 or, equivalently, A:=(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 Example T(2,1)=2 because we have udUdd and Uddud.
%e Triangle begins:
%e 1;
%e 1,1;
%e 7,2,1;
%e 47,15,3,1;
%e 361,108,24,4,1;
%p 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+z-t*z-z*A^2): Gserz:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 10 do P[n]:=sort(coeff(Gserz,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 # second Maple program:
%p b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,
%p `if`(x=0, 1, b(x-1, y-1, t)*`if`(t and y=1, z, 1)+
%p b(x-1, y+2, false)+b(x-2, y+1, is(y=0)))))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..n))(b(3*n, 0, false)):
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Oct 06 2015
%t b[x_, y_, t_] := b[x, y, t] = Expand[If[y < 0 || y > x, 0, If[x == 0, 1, b[x - 1, y - 1, t]*If[t && y == 1, z, 1] + b[x - 1, y + 2, False] + b[x - 2, y + 1, y == 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n}]][b[3*n, 0, False]]; Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 29 2016, after _Alois P. Heinz_ *)
%Y Cf. A027307, A108431, A108432, A108434.
%K nonn,tabl
%O 0,4
%A _Emeric Deutsch_, Jun 03 2005
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