cp's OEIS Frontend

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.

%I A088459 #25 Feb 28 2023 04:48:53
%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,
%T A088459 1,1,6,30,75,150,200,200,150,75,30,6,1,1,7,42,126,315,525,700,700,525,
%U A088459 315,126,42,7,1,1,8,56,196,588,1176,1960,2450,2450,1960,1176,588,196,56,8,1
%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_, Feb 22 2004
%C A088459 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 A088459 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 A088459 T(n, k) = binomial(n, ceiling(k/2))* binomial(n-1, 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 T(n, k) begins:
%e A088459 [1] 1,1,
%e A088459 [2] 1,2, 2,  1,
%e A088459 [3] 1,3, 6,  6,   3,   1,
%e A088459 [4] 1,4,12, 18,  18,  12,   4,   1,
%e A088459 [5] 1,5,20, 40,  60,  60,  40,  20,   5,   1,
%e A088459 [6] 1,6,30, 75, 150, 200, 200, 150,  75,  30,   6,  1,
%e A088459 [7] 1,7,42,126, 315, 525, 700, 700, 525, 315, 126, 42,    7,   1,
%e A088459 [8] 1,8,56,196, 588,1176,1960,2450,2450,1960,1176,588,  196,  56,  8, 1,
%e A088459 [9] 1,9,72,288,1008,2352,4704,7056,8820,8820,7056,4704,2352,1008,288,72,9,1
%p A088459 A088459 := proc(n,k)
%p A088459     binomial(n,ceil(k/2))*binomial(n-1,floor(k/2)) ;
%p A088459 end proc:
%p A088459 seq(seq(A088459(n,k),k=0..2*n-1),n=1..10) ; # _R. J. Mathar_, Apr 02 2017
%t A088459 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 A088459 Columns 0-5 are sequences A000012, A000027, A002378, A002411, A006011 and A004302.
%Y A088459 Cf. A000984 (row sums).
%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_, Nov 17 2003