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

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, A150500, A202814.

Cf. A001006, A126869, A302184.

Programs

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

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)

Extensions

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