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%I #7 Jun 06 2021 21:43:51
%S 1,1,2,1,8,4,1,18,36,8,1,32,144,128,16,1,50,400,800,400,32,1,72,900,
%T 3200,3600,1152,64,1,98,1764,9800,19600,14112,3136,128,1,128,3136,
%U 25088,78400,100352,50176,8192,256,1,162,5184,56448,254016,508032,451584,165888,20736,512
%N Delannoy paths counted by number of weak peaks.
%C 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.
%H G. C. Greubel, <a href="/A133214/b133214.txt">Rows n = 0..50 of the triangle, flattened</a>
%H See Example 3 in Robert A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
%F T(n, k) = 2^k binomial(n,k)^2.
%F 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)).
%F Row sums are the central Delannoy numbers A001850.
%e Table begins:
%e \ k.0...1....2....3....4....5
%e n\
%e 0 |.1
%e 1 |.1...2
%e 2 |.1...8....4
%e 3 |.1..18...36....8
%e 4 |.1..32..144..128...16
%e 5 |.1..50..400..800..400...32
%e T(2,1) = 8 counts the paths UUDD, UDDU, UHD, DUUD, DUDU, DUH, DHU, HDU
%e because each contains a single UD or a single H but not both.
%t Table[2^k*Binomial[n, k]^2, {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 06 2021 *)
%o (Sage) flatten([[2^k*binomial(n,k)^2 for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 06 2021
%Y Cf. A001850 (row sums).
%K nonn,tabl
%O 0,3
%A _David Callan_, Dec 18 2007
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