|
Search: id:A133214
|
|
|
| A133214 |
|
Delannoy paths counted by number of weak peaks. |
|
+0
1
|
|
| 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; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
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
|
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.
Adjacent sequences: A133211 A133212 A133213 this_sequence A133215 A133216 A133217
Sequence in context: A075647 A085470 A099379 this_sequence A110107 A110446 A109979
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
David Callan (callan(AT)stat.wisc.edu), Dec 18 2007
|
|
|
Search completed in 0.002 seconds
|
