A302184 Number of 3D walks of type abe.

1, 2, 7, 26, 108, 472, 2159, 10194, 49396, 244328, 1229308, 6273896, 32410096, 169181664, 891181607, 4731912082, 25302648644, 136150941064, 736747902236, 4007011320808, 21893702201648, 120125750018656, 661630546993116, 3656966382542984, 20278320788680912, 112782556853239712

Offset: 0

Author

N. J. A. Sloane, Apr 09 2018

Keywords

• nonn
• walk

Comments

See Dershowitz (2017) for precise definition.

Links

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

Crossrefs

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A126120, A138547, A145847, A145867, A150500, A202814.

Programs

Maple

a := n -> 2*add(binomial(n, k)*binomial(k, k/2)*binomial(2*(n-k), n-k)/(k+2), k = 0..n, 2): seq(a(n), n = 0..25);  # Peter Luschny, Nov 30 2024

Python

from math import comb as binomial
def a(n: int):
    return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(2*(n-k), n-k) for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Nov 30 2024

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A000984(n-k). - Mélika Tebni, Nov 30 2024

Extensions

a(12)-a(25) from Mélika Tebni, Nov 30 2024