A302187 Number of 3D walks of type bcc.

1, 2, 8, 30, 138, 620, 3060, 14910, 76650, 390852, 2063376, 10832052, 58264668, 312123240, 1702423008, 9256786110, 51036229530, 280696824980, 1560925457520, 8663089672380, 48512836025940, 271229902496280, 1527733861191720, 8593482390429300, 48642125421855420, 275014629509319000

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)*A001405(k)*A018224(n-k). - Mélika Tebni, Nov 25 2024

Conjecture D-finite with recurrence (n+1)*(n+2)^2*a(n) +2*(7*n^3-17*n-6)*a(n-1) -4*(4*n-1)*(4*n^2+9*n-12)*a(n-2) +8*(-70*n^3+330*n^2-461*n+225)*a(n-3) +48*(n-3)*(23*n^2-66*n+64)*a(n-4) +288*(7*n-17)*(n-3)*(n-4)*a(n-5) -3456*(n-5)*(n-3)*(n-4)*a(n-6)=0. - R. J. Mathar, Oct 29 2025

Prog

(Python)
from math import comb as binomial
def a(n):
    return sum(binomial(n, k)*binomial(k, k//2)*binomial(n-k, (n-k)//2)**2 for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Nov 25 2024

Crossrefs

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

Cf. A001405, A018224.

Sequence in context: A150791 A026693 A150792 * A266043 A358962 A265950

Adjacent sequences: A302184 A302185 A302186 * A302188 A302189 A302190

Keyword

nonn,walk

Author

N. J. A. Sloane, Apr 09 2018

Extensions

a(12)-a(25) from Nachum Dershowitz, Aug 03 2020

Status

approved