A302186 Number of 3D walks of type ace.

1, 3, 11, 44, 188, 842, 3911, 18692, 91412, 455540, 2306028, 11829424, 61375408, 321583108, 1699500055, 9049714852, 48513809796, 261638920412, 1418673379052, 7730011715760, 42305916178288, 232475082183544, 1282208011668988, 7096065370945168, 39394821683770960, 219341739839760912

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

Binomial transform of A145847. - Mélika Tebni, Nov 29 2024

Conjecture D-finite with recurrence (n+3)*(n+2)*a(n) +4*(-3*n^2-8*n-3)*a(n-1) +8*(4*n^2+3*n-3)*a(n-2) +48*(n-2)^2*a(n-3) +144*-(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Oct 29 2025

Prog

(Python)
from math import comb as binomial
def C(n): return (binomial(2*n, n)//(n+1)) # Catalan numbers
def row(n: int) -> list[int]:
     return sum(binomial(n, k)*sum(binomial(k, j)*C((j+1)//2)*C(j//2)*(2*(j//2)+1) for j in range(k+1)) for k in range(n+1))
for n in range(26): print(row(n)) # Mélika Tebni, Nov 29 2024

Crossrefs

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867 (number of 3D walks of type acd), A150500, A202814.

Sequence in context: A026887 A389410 A151106 * A151107 A063018 A293468

Adjacent sequences: A302183 A302184 A302185 * A302187 A302188 A302189

Keyword

nonn,walk

Author

N. J. A. Sloane, Apr 09 2018

Extensions

a(12)-a(25) from Mélika Tebni, Nov 29 2024

Status

approved