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A276418
Starting a random walk on Z at 0 triangle T(j,k) gives the number of paths of length 2*j returning to 0 exactly k times.
1
1, 2, 2, 6, 6, 4, 20, 20, 16, 8, 70, 70, 60, 40, 16, 252, 252, 224, 168, 96, 32, 924, 924, 840, 672, 448, 224, 64, 3432, 3432, 3168, 2640, 1920, 1152, 512, 128, 12870, 12870, 12012, 10296, 7920, 5280, 2880, 1152, 256, 48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512, 184756, 184756, 175032, 155584, 128128, 96096, 64064, 36608, 16896, 5632, 1024
OFFSET
0,2
COMMENTS
The number of paths of odd length 2*j+1 is the same as the number of even length 2*j (returning to 0 exactly k times).
LINKS
FORMULA
T(j,k) = (2^k)*C(2*j-k,j-k).
T(j,0) = T(j,1) for j>0.
T(j,0) = A000984(j).
T(j,1) = A000984(j) for j>0.
T(j,2) = A128650(j+1).
T(j,j) = A000079(j).
T(j,j-1) = A057711(j+1) for j>0.
EXAMPLE
Triangle T(j,k) begins:
1
2, 2
6, 6, 4
20, 20, 16, 8
70, 70, 60, 40, 16
252, 252, 224, 168, 96, 32
924, 924, 840, 672, 448, 224, 64
3432, 3432, 3168, 2640, 1920, 1152, 512, 128
12870, 12870, 12012, 10296, 7920, 5280, 2880, 1152, 256
48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512
PROG
(GAP) Flat(List([0..10], j->List([0..j], k->2^k*Binomial(2*j-k, j-k)))); # Muniru A Asiru, May 18 2018
CROSSREFS
Sequence in context: A033730 A033754 A171661 * A173869 A247377 A167909
KEYWORD
nonn,tabl,walk,easy
AUTHOR
Franz Vrabec, Sep 27 2016
STATUS
approved