%I A108450
%S A108450 2,10,58,402,3122,26010,227050,2049186,18964194,178976426,1715905050,
%T A108450 16665027378,163611970066,1621103006010,16189480081354,162791835045698,
%U A108450 1646810150270914,16748008972020554,171135004105459194
%N A108450 Number of pyramids 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) (a pyramid is a sequence u^pd^p 
               or U^pd^(2p) for some positive integer p, starting at the x-axis).
%C A108450 A108450(n)=sum(k*A108445(k),k=1..n) (for example, A108450(3)=1*18+2*8+3*8=58). 
               A108450(n)=2*A108453(n). A108450 =2*partial sums of A032349.
%D A108450 Problem 10658, American Math. Monthly, 107, 2000, 368-370.
%F A108450 G.f.=2zA^2/(1-z), 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 A108450 a(2)=10 because in the A027307(2)=10 paths we have alltogether 10 pyramids 
               (shown between parentheses): (ud)(ud), (ud)(Udd), (uudd), uUddd, 
               (Udd)(ud), (Udd)(Udd), Ududd, UdUddd, Uuddd, (UUdddd).
%p A108450 A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/
               3: g:=2*z*A^2/(1-z): gser:=series(g,z=0,25): seq(coeff(gser,z^n),
               n=1..22);
%Y A108450 Cf. A027307, A108445, A108453, A032349.
%Y A108450 Sequence in context: A075870 A074608 A086871 this_sequence A112369 A124964 
               A026132
%Y A108450 Adjacent sequences: A108447 A108448 A108449 this_sequence A108451 A108452 
               A108453
%K A108450 nonn
%O A108450 1,1
%A A108450 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 11 2005