A302180 Number of 3D walks of type aad.

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

Author

N. J. A. Sloane, Apr 09 2018

Keywords

• nonn
• walk

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.

Crossrefs

Cf. A000108, A000984, A001006, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Programs

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

Extensions

a(14)-a(26) from Farzan Byramji, Mar 06 2021