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%I #14 Nov 27 2024 18:55:00
%S 1,1,5,12,62,200,1065,3990,21714,89082,492366,2147376,12004740,
%T 54718092,308559537,1454116950,8255788970,39935276810,227976044010,
%U 1126178350440,6457854821340,32456552441040,186814834574550,952569927106980,5500292590186380,28391993275117500
%N Number of 3D walks of type abc.
%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 From _Mélika Tebni_, Nov 27 2024: (Start)
%F a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A018224(n-k).
%F a(2*n+1) = A135394(n) / (2*n+2).
%F a(2*n) = A302181(n). (End)
%o (Python)
%o from math import comb as binomial
%o def row(n: int) -> list[int]:
%o return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(n-k, (n-k)//2)**2 for k in range(n+1))
%o for n in range(26): print(row(n)) # _Mélika Tebni_, Nov 27 2024
%Y Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.
%Y Cf. A018224, A126120, A135394, A302181.
%K nonn,walk
%O 0,3
%A _N. J. A. Sloane_, Apr 09 2018
%E a(13)-a(25) from _Mélika Tebni_, Nov 27 2024