%I A088459
%S A088459 1,1,1,2,2,1,1,3,6,6,3,1,1,4,12,18,18,12,4,1,1,5,20,40,60,60,40,20,5,1,
%T A088459 1,6,30,75,150,200,200,150,75,30,6,1,1,7,42,126,315,525,700,700,525,315,
%U A088459 126,42,7,1,1,8,56,196,588,1176,1960,2450,2450,1960,1176,588,196,56,8,
               1
%N A088459 Triangle read by rows: T(n,k) represents the number of lozenge tilings 
               of an (n,1,n)-hexagon which include the non-vertical tile above the 
               main diagonal starting in position k+1.
%C A088459 Rows are of length 2, 4, 6, 8, 10, 12, ...
%C A088459 T(n,k)= number of symmetric Dyck paths of length 4n and having k peaks. 
               Example: T(2,3)=2 because we have UU*DU*DU*DD and U*DUU*DDU*D, where 
               U=(1,1), D=(1,-1) and * shows the peaks. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Feb 22 2004
%F A088459 T(n, k) = (n choose ceiling(k/2))*(n-1 choose floor(k/2)), n>0 and k=0 
               to 2n-1.
%e A088459 For example, the number of tilings of a 4,1,4 hexagon which includes 
               the non-vertical tile above the main diagonal starting in position 
               3 is T(4,2)=12.
%e A088459 Triangle begins:
%e A088459 1,1,
%e A088459 1,2,2,1,
%e A088459 1,3,6,6,3,1,
%e A088459 1,4,12,18,18,12,4,1,
%e A088459 1,5,20,40,60,60,40,20,5,1,
%e A088459 1,6,30,75,150,200,200,150,75,30,6,1,
%e A088459 1,7,42,126,315,525,700,700,525,315,126,42,7,1,
%e A088459 1,8,56,196,588,1176,1960,2450,2450,1960,1176,588,196,56,8,1,
%e A088459 1,9,72,288,1008,2352,4704,7056,8820,8820,7056,4704,2352,1008,288,72,9,
               1
%Y A088459 Columns 0-5 are sequences A000012, A000027, A002378, A002411, A006011 
               and A004302.
%Y A088459 Sequence in context: A156133 A010048 A055870 this_sequence A007799 A122888 
               A092113
%Y A088459 Adjacent sequences: A088456 A088457 A088458 this_sequence A088460 A088461 
               A088462
%K A088459 easy,nonn,tabf
%O A088459 1,4
%A A088459 Christopher Hanusa (chanusa(AT)washington.edu), Nov 14 2003
%E A088459 Edited and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net) Nov 
               17 2003