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.

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, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 7, 42, 126, 315, 525, 700, 700, 525, 315, 126, 42, 7, 1, 1, 8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1

Offset: 1

Author

Christopher Hanusa (chanusa(AT)washington.edu), Nov 14 2003

Keywords

• easy
• nonn
• tabf

Comments

Rows are of length 2, 4, 6, 8, 10, 12, ...

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

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

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

Examples

			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.
Triangle T(n, k) begins:
[1] 1,1,
[2] 1,2, 2,  1,
[3] 1,3, 6,  6,   3,   1,
[4] 1,4,12, 18,  18,  12,   4,   1,
[5] 1,5,20, 40,  60,  60,  40,  20,   5,   1,
[6] 1,6,30, 75, 150, 200, 200, 150,  75,  30,   6,  1,
[7] 1,7,42,126, 315, 525, 700, 700, 525, 315, 126, 42,    7,   1,
[8] 1,8,56,196, 588,1176,1960,2450,2450,1960,1176,588,  196,  56,  8, 1,
[9] 1,9,72,288,1008,2352,4704,7056,8820,8820,7056,4704,2352,1008,288,72,9,1
		

Crossrefs

Columns 0-5 are sequences A000012, A000027, A002378, A002411, A006011 and A004302.

Cf. A000984 (row sums).

Programs

Maple

A088459 := proc(n,k)
    binomial(n,ceil(k/2))*binomial(n-1,floor(k/2)) ;
end proc:
seq(seq(A088459(n,k),k=0..2*n-1),n=1..10) ; # R. J. Mathar, Apr 02 2017

Mathematica

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 *)

Formula

T(n, k) = binomial(n, ceiling(k/2))* binomial(n-1, floor(k/2)), n>0 and k=0 to 2n-1.

Extensions

Edited and extended by Ray Chandler, Nov 17 2003