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A302180
Number of 3D walks of type aad.
1
1, 1, 3, 7, 23, 71, 251, 883, 3305, 12505, 48895, 193755, 783355, 3205931, 13302329, 55764413, 236174933, 1008773269, 4343533967, 18834033443, 82201462251, 360883031291, 1592993944723, 7066748314147, 31493800133173, 140953938878821, 633354801073571, 2856369029213263
OFFSET
0,3
COMMENTS
See Dershowitz (2017) for precise definition.
Number of 3D walks of length n in the first octant using steps (1, 1, 0), (1, -1, 0), (1, 0, 1), (1, 0, -1) and (1, 0, 0) that start at the origin and end at (n, 0, 0). The analogous problem in 2D is given by the Motzkin numbers A001006. - Farzan Byramji, Mar 06 2021
Inverse binomial transform of A145867 (Number of 3D walks of type aae). - Mélika Tebni, Nov 05 2024
LINKS
Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
MAPLE
M := n-> add(binomial(n, 2*k)*binomial(2*k, k)/(k+1), k = 0 .. iquo(n, 2)): # Motzkin numbers
A302180 := n-> add((-1)^(n-k)*binomial(n, k)*add(binomial(k, j)*M(j)*M(k-j), j=0..k), k=0..n): seq(A302180(n), n = 0 .. 26); # Mélika Tebni, Nov 05 2024
KEYWORD
nonn,walk,changed
AUTHOR
N. J. A. Sloane, Apr 09 2018
EXTENSIONS
a(14)-a(26) from Farzan Byramji, Mar 06 2021
STATUS
approved