cp's OEIS Frontend

A302185 Number of 3D n-step walks of type acc.

%I A302185 #28 Apr 20 2025 03:57:03
%S A302185 1,2,7,24,98,400,1785,7980,37674,178164,874146,4294752,21667932,
%T A302185 109436184,563910633,2908233900,15235550330,79870553620,424021948350,
%U A302185 2252356700880,12088746573540,64913104882080,351594254659830,1905139854213960,10399223643879420,56783986550235000
%N A302185 Number of 3D n-step walks of type acc.
%C A302185 See Dershowitz (2017) for precise definition.
%H A302185 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 A302185 From _Mélika Tebni_, Dec 06 2024: (Start)
%F A302185 E.g.f.: (BesselI(0, 2*x) + BesselI(1, 2*x))^2*BesselI(1, 2*x) / x.
%F A302185 a(n) = Sum_{k=0..n} binomial(n, k)*A005558(k)*A001405(n-k).
%F A302185 a(2*n+1) = 2*A302182(2*n+1) = A135394(n) / (n+1).
%F A302185 For n > 0, a(A000918(n)) is odd. (End)
%p A302185 b:= n-> binomial(n, floor(n/2))*binomial(n+1, floor((n+1)/2)):
%p A302185 C:= n-> binomial(2*n, n)/(n+1):
%p A302185 a:= n-> add(binomial(n, 2*k)*C(k)*b(n-2*k), k=0..n/2):
%p A302185 seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 06 2024
%p A302185 # second Maple program:
%p A302185 a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 24][n+1],
%p A302185       (8*(14*n^4+85*n^3+190*n^2+188*n+63)*a(n-1)+4*(n-1)*
%p A302185       (80*n^4+418*n^3+676*n^2+269*n-108)*a(n-2)-96*(n-1)*(n-2)*
%p A302185       (10*n^2+31*n+27)*a(n-3)-144*(n-1)*(n-2)*(n-3)*(8*n^2+33*n+36)*
%p A302185        a(n-4))/((n+4)*(n+3)*(n+2)*(8*n^2+17*n+11)))
%p A302185     end:
%p A302185 seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 06 2024
%t A302185 b[n_] := Binomial[n, Floor[n/2]]*Binomial[n+1, Floor[(n+1)/2]];
%t A302185 c[n_] := Binomial[2*n, n]/(n+1);
%t A302185 a[n_] := Sum[Binomial[n, 2*k]*c[k]*b[n - 2*k], {k, 0, n/2}];
%t A302185 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 28 2025, after _Alois P. Heinz_ *)
%o A302185 (Python)
%o A302185 from math import comb as binomial
%o A302185 def C(n): return (binomial(2*n, n)//(n+1)) # Catalan numbers
%o A302185 def a(n):
%o A302185     return sum(binomial(n, k)*C((k+1)//2)*C(k//2)*(2*(k//2)+1)*binomial(n-k, (n-k)//2) for k in range(n+1))
%o A302185 print([a(n) for n in range(26)]) # _Mélika Tebni_, Dec 06 2024
%Y A302185 Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.
%Y A302185 Cf. A000918, A001405, A005558, A005566, A135394, A302182.
%K A302185 nonn,walk
%O A302185 0,2
%A A302185 _N. J. A. Sloane_, Apr 09 2018
%E A302185 a(13)-a(25) from _Mélika Tebni_, Dec 06 2024