OFFSET
0,3
COMMENTS
T(n,k) = number of Delannoy paths (A001850) of size n with k weak peaks. A (central) Delannoy path is a lattice path of upsteps U=(1,1), downsteps D=(1,-1) and horizontal steps H=(2,0) that starts at the origin and ends on the x-axis. Its size is #Us + #Hs. Thus a Delannoy path of size n ends at the point (2n,0). A weak peak is a UD or an H.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
See Example 3 in Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
FORMULA
T(n, k) = 2^k binomial(n,k)^2.
G.f.: Sum_{n>=k>=0} T(n,k) x^n y^k = 1/Sqrt((1-x)^2 - 4*x*y*(1+x-x*y)).
Row sums are the central Delannoy numbers A001850.
EXAMPLE
Table begins:
\ k.0...1....2....3....4....5
n\
0 |.1
1 |.1...2
2 |.1...8....4
3 |.1..18...36....8
4 |.1..32..144..128...16
5 |.1..50..400..800..400...32
T(2,1) = 8 counts the paths UUDD, UDDU, UHD, DUUD, DUDU, DUH, DHU, HDU
because each contains a single UD or a single H but not both.
MATHEMATICA
Table[2^k*Binomial[n, k]^2, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 06 2021 *)
PROG
(Sage) flatten([[2^k*binomial(n, k)^2 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 06 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
David Callan, Dec 18 2007
STATUS
approved