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A378849 a(n) is the total number of paths starting at (0,0), ending at (n,0), consisting of steps (1,1), (1,0), (1,-2), and staying on or above y = -1.

%I A378849 #18 Dec 18 2024 22:37:16
%S A378849 1,1,1,3,9,21,48,120,309,787,2011,5215,13652,35894,94823,251889,
%T A378849 672285,1801185,4842757,13064059,35349463,95912989,260896318,
%U A378849 711338596,1943690464,5321704006,14597781706,40112702176,110404515703,304338523999,840140172621,2322386563353
%N A378849 a(n) is the total number of paths starting at (0,0), ending at (n,0), consisting of steps (1,1), (1,0), (1,-2), and staying on or above y = -1.
%F A378849 a(n) = hypergeom([4/3, (1-n)/3, (2-n)/3, -n/3], [1/3, 3/2, 2], -27/4). - _Peter Luschny_, Dec 18 2024
%p A378849 a:= proc(n) option remember; `if`(n<4, [1$3, 3][n+1],
%p A378849       (2*(8*n^3+3*n^2-25*n-6)*a(n-1)-2*(n-1)*(12*n^2-9*n-10)*
%p A378849        a(n-2)+(43*n+13)*(n-1)*(n-2)*a(n-3)-31*(n-1)*(n-2)*
%p A378849         (n-3)*a(n-4))/(2*(2*n+3)*(n+3)*(n-2)))
%p A378849     end:
%p A378849 seq(a(n), n=0..31);  # _Alois P. Heinz_, Dec 09 2024
%o A378849 (PARI) a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+1, k)/(k+1)) \\ _Thomas Scheuerle_, Dec 09 2024
%Y A378849 Cf. A071879, A116411, A378850.
%K A378849 nonn
%O A378849 0,4
%A A378849 _Emely Hanna Li Lobnig_, Dec 09 2024
%E A378849 More terms from _Alois P. Heinz_, Dec 09 2024