This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378850 #16 Dec 18 2024 22:36:56 %S A378850 1,1,1,4,13,31,73,190,505,1316,3431,9065,24122,64325,172082,462344, %T A378850 1246685,3371135,9140289,24847422,67708743,184906614,505986933, %U A378850 1387240401,3810083424,10481797131,28880894706,79692785251,220203155689,609242057143,1687666776031 %N A378850 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 A378850 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 A378850 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 A378850 seq(simplify(a(n)), n = 0..30); # _Peter Luschny_, Dec 18 2024 %o A378850 (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 A378850 Cf. A378849, A116411, A071879, A026325. %K A378850 nonn %O A378850 0,4 %A A378850 _Emely Hanna Li Lobnig_, Dec 09 2024