A302183 Number of 3D n-step walks of type abd.

1, 1, 4, 10, 39, 131, 521, 1989, 8149, 33205, 139870, 592120, 2552155, 11079303, 48639722, 214997228, 957817013, 4292316197, 19349957108, 87663905954, 399038606291, 1823961268751, 8369603968599, 38540835938335, 178056111047329, 825079806039121, 3833960405339446

Offset

0,3

Comments

See Dershowitz (2017) for precise definition.

Links

Table of n, a(n) for n=0..26.

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

Formula

From Mélika Tebni, Dec 03 2024: (Start)

a(n) = Sum_{k=0..n} binomial(n, k)*A126869(k)*A001006(n-k).

Inverse binomial transform of A302184. (End)

Prog

(Python)
from math import comb as binomial
def M(n): return sum(binomial(n, 2*k)*binomial(2*k, k)//(k+1) for k in range(n//2+1)) # Motzkin numbers
def a(n):
    return sum(binomial(n, k)*binomial(k, k//2)*((k+1) %2)*M(n-k) for k in range(n+1))
print([a(n) for n in range(27)]) # Mélika Tebni, Dec 03 2024

Crossrefs

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

Cf. A001006, A126869, A302184.

Sequence in context: A320806 A149202 A151447 * A149203 A038168 A186337

Adjacent sequences: A302180 A302181 A302182 * A302184 A302185 A302186

Keyword

nonn,walk,changed

Author

N. J. A. Sloane, Apr 09 2018

Extensions

a(13)-a(26) from Mélika Tebni, Dec 03 2024

Status

approved