A378850 - OEIS

%I #16 Dec 18 2024 22:36:56

%S 1,1,1,4,13,31,73,190,505,1316,3431,9065,24122,64325,172082,462344,

%T 1246685,3371135,9140289,24847422,67708743,184906614,505986933,

%U 1387240401,3810083424,10481797131,28880894706,79692785251,220203155689,609242057143,1687666776031

%N 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 = -2.

%e For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0).

%p a := n -> hypergeom([4/3, 5/3, (1-n)/3, (2-n)/3, -n/3], [1/3, 2/3, 5/2, 2], -27/4):

%p seq(simplify(a(n)), n = 0..30);  # _Peter Luschny_, Dec 18 2024

%o (PARI) a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+3, k+1)/(2*k+3)) \\ _Thomas Scheuerle_, Dec 09 2024

%Y Cf. A378849, A116411, A071879, A026325.

%K nonn

%O 0,4

%A _Emely Hanna Li Lobnig_, Dec 09 2024