%I A108444
%S A108444 5,73,857,9505,103341,1114969,11996209,128989249,1387480981,14937170089,
%T A108444 160978217225,1736820843233,18760031574077,202856430706617,
%U A108444 2195832009812065,23792481053343361,258038743598973477
%N A108444 Number of triple descents (i.e. ddd's) in all 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).
%C A108444 a(n)=sum(k*A108443(n,k),k=1..2n-1). Example: a(3)=1*24+2*15+3*3+4*1=73.
%D A108444 Problem 10658, American Math. Monthly, 107, 2000, 368-370.
%F A108444 G.f.=zA(2A^2-2zA^2-zA-2)/(1-2zA-3zA^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 A108444 a(2)=5 because in the ten paths udud, udUdd, uudd, uU(ddd), Uddud, UddUdd, 
               Ududd, UdU(ddd), Uu(ddd) and UU(d[dd)d] (see A027307) we have 5 ddd's 
               (shown between parentheses).
%p A108444 A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/
               3: G:=z*A*(-z*A-2*z*A^2-2+2*A^2)/(1-3*z*A^2-2*z*A): Gser:=series(G,
               z=0,26): seq(coeff(Gser,z^n),n=2..21);
%Y A108444 Cf. A027307, A108443.
%Y A108444 Sequence in context: A070530 A059017 A099667 this_sequence A155662 A159509 
               A127167
%Y A108444 Adjacent sequences: A108441 A108442 A108443 this_sequence A108445 A108446 
               A108447
%K A108444 nonn
%O A108444 2,1
%A A108444 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 10 2005