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A133214
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Delannoy paths counted by number of weak peaks.
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1
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1, 1, 2, 1, 8, 4, 1, 18, 36, 8, 1, 32, 144, 128, 16, 1, 50, 400, 800, 400, 32, 1, 72, 900, 3200, 3600, 1152, 64, 1, 98, 1764, 9800, 19600, 14112, 3136, 128, 1, 128, 3136, 25088, 78400, 100352, 50176, 8192, 256, 1, 162, 5184, 56448, 254016, 508032, 451584, 165888, 20736, 512
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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Table[2^k*Binomial[n, k]^2, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 06 2021 *)
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PROG
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(Sage) flatten([[2^k*binomial(n, k)^2 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 06 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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