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

0,2

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

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

Conjecture D-finite with recurrence (n+2)^2*a(n) +2*(-3*n^2-5*n-1)*a(n-1) -4*(n+3)*(n-1)*a(n-2) +24*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Oct 29 2025

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

Prog

(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

Crossrefs

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

Sequence in context: A150575 A150576 A150577 * A150578 A150579 A093638

Adjacent sequences: A302181 A302182 A302183 * A302185 A302186 A302187

Keyword

nonn,walk

Author

N. J. A. Sloane, Apr 09 2018

Extensions

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

Status

approved