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Number of 3D walks of type abe.
1

%I #19 Dec 01 2024 06:00:32

%S 1,2,7,26,108,472,2159,10194,49396,244328,1229308,6273896,32410096,

%T 169181664,891181607,4731912082,25302648644,136150941064,736747902236,

%U 4007011320808,21893702201648,120125750018656,661630546993116,3656966382542984,20278320788680912,112782556853239712

%N Number of 3D walks of type abe.

%C See Dershowitz (2017) for precise definition.

%H Nachum Dershowitz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Dershowitz/dersh3.html">Touchard’s Drunkard</a>, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

%F a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A000984(n-k). - _Mélika Tebni_, Nov 30 2024

%p a := n -> 2*add(binomial(n, k)*binomial(k, k/2)*binomial(2*(n-k), n-k)/(k+2), k = 0..n, 2): seq(a(n), n = 0..25); # _Peter Luschny_, Nov 30 2024

%o (Python)

%o from math import comb as binomial

%o def a(n: int):

%o return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(2*(n-k), n-k) for k in range(n+1))

%o print([a(n) for n in range(26)]) # _Mélika Tebni_, Nov 30 2024

%Y Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A126120, A138547, A145847, A145867, A150500, A202814.

%K nonn,walk

%O 0,2

%A _N. J. A. Sloane_, Apr 09 2018

%E a(12)-a(25) from _Mélika Tebni_, Nov 30 2024