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%I #25 Feb 28 2023 04:48:53
%S 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,
%T 1,1,6,30,75,150,200,200,150,75,30,6,1,1,7,42,126,315,525,700,700,525,
%U 315,126,42,7,1,1,8,56,196,588,1176,1960,2450,2450,1960,1176,588,196,56,8,1
%N 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 Rows are of length 2, 4, 6, 8, 10, 12, ...
%C 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_, Feb 22 2004
%C T(n,k) is also the number of nodes at distance k from a specified node in the n-odd graph for k in 1..n-1. - _Eric W. Weisstein_, Mar 23 2018
%C T(n,k) seems to be the k-th Lie-Betti number of the star graph on n vertices. See A360571 for additional information and references. - _Samuel J. Bevins_, Feb 12 2023
%F T(n, k) = binomial(n, ceiling(k/2))* binomial(n-1, floor(k/2)), n>0 and k=0 to 2n-1.
%e 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 Triangle T(n, k) begins:
%e [1] 1,1,
%e [2] 1,2, 2, 1,
%e [3] 1,3, 6, 6, 3, 1,
%e [4] 1,4,12, 18, 18, 12, 4, 1,
%e [5] 1,5,20, 40, 60, 60, 40, 20, 5, 1,
%e [6] 1,6,30, 75, 150, 200, 200, 150, 75, 30, 6, 1,
%e [7] 1,7,42,126, 315, 525, 700, 700, 525, 315, 126, 42, 7, 1,
%e [8] 1,8,56,196, 588,1176,1960,2450,2450,1960,1176,588, 196, 56, 8, 1,
%e [9] 1,9,72,288,1008,2352,4704,7056,8820,8820,7056,4704,2352,1008,288,72,9,1
%p A088459 := proc(n,k)
%p binomial(n,ceil(k/2))*binomial(n-1,floor(k/2)) ;
%p end proc:
%p seq(seq(A088459(n,k),k=0..2*n-1),n=1..10) ; # _R. J. Mathar_, Apr 02 2017
%t Table[Binomial[n, Ceiling[k/2]] Binomial[n - 1, Floor[k/2]], {n, 10}, {k, 0, 2 n - 1}] // Flatten (* _Eric W. Weisstein_, Mar 23 2018 *)
%Y Columns 0-5 are sequences A000012, A000027, A002378, A002411, A006011 and A004302.
%Y Cf. A000984 (row sums).
%K easy,nonn,tabf
%O 1,4
%A Christopher Hanusa (chanusa(AT)washington.edu), Nov 14 2003
%E Edited and extended by _Ray Chandler_, Nov 17 2003
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