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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

LINKS

Table of n, a(n) for n=1..72.

FORMULA

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

EXAMPLE

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 begins:

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,

1,9,72,288,1008,2352,4704,7056,8820,8820,7056,4704,2352,1008,288,72,9,1

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 *)

CROSSREFS

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

Cf. A000984 (row sums).

Sequence in context: A156133 A010048 A055870 * A300699 A007799 A122888

Adjacent sequences:  A088456 A088457 A088458 * A088460 A088461 A088462

KEYWORD

easy,nonn,tabf

AUTHOR

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

EXTENSIONS

Edited and extended by Ray Chandler, Nov 17 2003

STATUS

approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)