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 A361728 #15 Mar 17 2024 08:41:58
%S A361728 1,6,108,2238,51126,1234836,30933846,795124008,20832161238,
%T A361728 553908550416,14901620938668,404737904238768,11080360585597974,
%U A361728 305375448989901564,8464333256181647028,235772833122673888788,6595763835075158604618
%N A361728 Diagonal of rational function 1/(1 - (1 + x*y*z) * (x + y + z)).
%F A361728 a(n) = Sum_{k=0..n} (3*k)!/k!^3 * binomial(3*k,n-k).
%F A361728 From _Vaclav Kotesovec_, Mar 22 2023: (Start)
%F A361728 Recurrence: (n-1)*n^2*a(n) = 2*(n-1)*(13*n^2 - 13*n + 3)*a(n-1) + 12*(9*n^3 - 27*n^2 + 23*n - 3)*a(n-2) + 18*(9*n^3 - 36*n^2 + 38*n - 3)*a(n-3) + 12*(9*n^3 - 45*n^2 + 56*n - 2)*a(n-4) + 3*n*(3*n - 11)*(3*n - 7)*a(n-5).
%F A361728 a(n) ~ c * d^n / n, where d = 29.8094342438507627973286122946283855557156321402886102401458498265933891... is the real root of the equation -27 - 81*d - 81*d^2 - 27*d^3 + d^4 = 0 and c = sqrt(3)/(2*Pi) = 0.27566444771089602475566324915648472069869324018332... (End)
%t A361728 Table[Sum[(3*k)!/k!^3 * Binomial[3*k,n-k], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 22 2023 *)
%o A361728 (PARI) a(n) = sum(k=0, n, (3*k)!/k!^3*binomial(3*k, n-k));
%Y A361728 Cf. A361729, A361730.
%K A361728 nonn
%O A361728 0,2
%A A361728 _Seiichi Manyama_, Mar 22 2023