%I A108451
%S A108451 1,1,1,6,3,1,44,16,5,1,344,116,30,7,1,2856,928,224,48,9,1,24816,7856,
%T A108451 1840,376,70,11,1,223016,69264,15912,3184,580,96,13,1,2056256,629472,
%U A108451 142592,28176,5080,844,126,15,1,19344472,5855472,1312360,256992,46072
%N A108451 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 pyramids 
               of the first kind (a pyramid of the first kind is a sequence u^pd^p 
               for some positive integer p, starting at the x-axis).
%C A108451 Also 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 pyramids of the second kind (a pyramid 
               of the second kind is a sequence U^pd^(2p) for some positive integer 
               p, starting at the x-axis). Row sums yield A027307. Column 0 yields 
               A108452. Number of pyramids of the first kind in all paths from (0,
               0) to (3n,0) is given by A108453.
%D A108451 Problem 10658, American Math. Monthly, 107, 2000, 368-370.
%F A108451 G.f. =(1-z)/[1-tz-z(1-z)A(1+A)], 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 A108451 T(2,1)=3 because we have (ud)Udd, (uudd) and Udd(ud), the pyramids of 
               the first kind being shown between parentheses.
%e A108451 Triangle begins:
%e A108451 1;
%e A108451 1,1;
%e A108451 6,3,1;
%e A108451 44,16,5,1;
%p A108451 A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/
               3: G:=(1-z)/(1-t*z-z*(1-z)*A*(1+A)): Gser:=simplify(series(G,z=0,
               13)): P[0]:=1: for n from 1 to 10 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 A108451 Cf. A027307, A108452, A108453, A108445.
%Y A108451 Sequence in context: A066717 A154969 A119743 this_sequence A122178 A126445 
               A033326
%Y A108451 Adjacent sequences: A108448 A108449 A108450 this_sequence A108452 A108453 
               A108454
%K A108451 nonn,tabl
%O A108451 0,4
%A A108451 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 11 2005