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 A071969 #25 Sep 27 2023 10:07:00
%S A071969 1,1,2,6,19,63,219,787,2897,10869,41414,159822,623391,2453727,9733866,
%T A071969 38877318,156206233,630947421,2560537092,10435207116,42689715279,
%U A071969 175243923783,721649457417,2980276087005,12340456995177,51222441676513,213090270498764,888321276659112
%N A071969 a(n) = Sum_{k=0..floor(n/3)} (binomial(n+1, k)*binomial(2*n-3*k, n-3*k)/(n+1)).
%C A071969 Diagonal of A071946. - _Emeric Deutsch_, Dec 15 2004
%C A071969 Last (largest) number of each row of A071946. - _David Scambler_, May 15 2012
%H A071969 Seiichi Manyama, <a href="/A071969/b071969.txt">Table of n, a(n) for n = 0..1000</a>
%H A071969 D. Merlini et al., <a href="https://doi.org/10.1016/S0166-218X(98)00126-7">Underdiagonal lattice paths with unrestricted steps</a>, Discrete Appl. Math., 91 (1999), 197-213 (d_n page 209).
%F A071969 G.f. (offset 1) is series reversion of (x-x^2)/(1+x^3).
%p A071969 A071969 := n->add( binomial(n+1,k)*binomial(2*n-3*k,n-3*k)/(n+1),k=0..floor(n/3));
%p A071969 Order:=30: g:=solve(series((H-H^2)/(1+H^3),H)=z,H): seq(coeff(g,z^n),n=1..28); # _Emeric Deutsch_, Dec 15 2004
%t A071969 Table[Sum[Binomial[n+1,k] Binomial[2n-3k,n-3k]/(n+1),{k,0,Floor[n/3]}],{n,0,40}] (* _Harvey P. Dale_, Jul 20 2022 *)
%o A071969 (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x^3)+x^2*O(x^n)),n+1))
%Y A071969 Cf. A071946 is the triangle and A119254 has the row sums.
%Y A071969 Cf. A006318, A052709, A365268, A366025.
%K A071969 nonn
%O A071969 0,3
%A A071969 _N. J. A. Sloane_, Jun 17 2002