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

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, A138547, A145847, A145867 (number of 3D walks of type acd), A150500, A202814.

Programs

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

Formula

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

Extensions

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