%I A108440
%S A108440 1,1,1,5,4,1,33,25,7,1,249,184,54,10,1,2033,1481,446,92,13,1,17485,
%T A108440 12620,3863,846,139,16,1,156033,111889,34637,7881,1411,195,19,1,1431281,
%U A108440 1021424,318812,74492,14102,2168,260,22,1,13412193,9536113,2995228
%N A108440 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=(2,
               1) steps among the steps leading to the first d step.
%D A108440 Problem 10658, American Math. Monthly, 107, 2000, 368-370.
%F A108440 G.f.=G=G(t, z)=1/(1-tzA-zA^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 A108440 T(2,1)=4 because we have udud, udUdd, uUddd and Uuddd.
%e A108440 Triangle begins:
%e A108440 .1;
%e A108440 .1,1;
%e A108440 .5,4,1;
%e A108440 .33,25,7,1;
%e A108440 .249,184,54,10,1;
%p A108440 A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/
               3: G:=1/(1-t*z*A-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
%Y A108440 Row sums yield A027307. Column 0 yields A034015.
%Y A108440 Cf. A027307, A034015, A108441.
%Y A108440 Sequence in context: A098494 A008955 A152862 this_sequence A102220 A109430 
               A085917
%Y A108440 Adjacent sequences: A108437 A108438 A108439 this_sequence A108441 A108442 
               A108443
%K A108440 nonn,tabl
%O A108440 0,4
%A A108440 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 08 2005