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A133214 Delannoy paths counted by number of weak peaks. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..51.

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. GF: Sum_{n>=k>=0}T(n,k) x^n y^k = 1/Sqrt[(1-x)^2 - 4*x*y*(1+x-x*y)]

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.

CROSSREFS

Row sums are the central Delannoy numbers A001850.

Sequence in context: A099379 A234014 A208931 * A191935 A142075 A156365

Adjacent sequences:  A133211 A133212 A133213 * A133215 A133216 A133217

KEYWORD

nonn,tabl

AUTHOR

David Callan, Dec 18 2007

STATUS

approved

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Last modified December 10 15:09 EST 2019. Contains 329896 sequences. (Running on oeis4.)