A302182 Number of 3D walks of type abc.

1, 1, 5, 12, 62, 200, 1065, 3990, 21714, 89082, 492366, 2147376, 12004740, 54718092, 308559537, 1454116950, 8255788970, 39935276810, 227976044010, 1126178350440, 6457854821340, 32456552441040, 186814834574550, 952569927106980, 5500292590186380, 28391993275117500

Offset

0,3

Comments

See Dershowitz (2017) for precise definition.

Links

Table of n, a(n) for n=0..25.

Nachum Dershowitz, Touchard’s Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Formula

From Mélika Tebni, Nov 27 2024: (Start)

a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A018224(n-k).

a(2*n+1) = A135394(n) / (2*n+2).

a(2*n) = A302181(n). (End)

Prog

(Python)
from math import comb as binomial
def row(n: int) -> list[int]:
    return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(n-k, (n-k)//2)**2 for k in range(n+1))
for n in range(26): print(row(n)) # Mélika Tebni, Nov 27 2024

Crossrefs

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

Cf. A018224, A126120, A135394, A302181.

Sequence in context: A346746 A111904 A015784 * A363452 A129723 A219770

Adjacent sequences: A302179 A302180 A302181 * A302183 A302184 A302185

Keyword

nonn,walk

Author

N. J. A. Sloane, Apr 09 2018

Extensions

a(13)-a(25) from Mélika Tebni, Nov 27 2024

Status

approved