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 A354157 #18 Apr 14 2023 13:44:25
%S A354157 1,1,5,104,836,7315,202895,1949900,19284511,1754890501,18058389349,
%T A354157 188502545504,5973492827120,63732573470888,685813307216632,
%U A354157 22303841469480032,243350841747362492,2670252449037801100,265034693078133749180,2936064912067020698720
%N A354157 Numerator of generalized Catalan number c_3(n) (see Comments).
%C A354157 c_3(n) = (1/3)*(1/(n+1/3))*(Product_{i=0..n-1}(n+i+1/3))/n!. The denominators are powers of 3.
%C A354157 If 1/3 is everywhere changed to 1 we get the usual Catalan numbers A000108.
%D A354157 J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.6.
%e A354157 The first few c_3(n) are 1, 1/3, 5/9, 104/81, 836/243, 7315/729, 202895/6561, 1949900/19683, 19284511/59049, 1754890501/1594323, 18058389349/4782969, ...
%p A354157 c := proc(n) 1/3 * 1/(n+1/3) * mul(n + i + 1/3, i = 0..(n-1))/n!: end;
%t A354157 c3[n_] := With[{k = 3}, Pochhammer[n+1+1/k, n-1]/(k*n!)];
%t A354157 Table[Numerator[c3[n]], {n, 1, 19}] (* _Jean-François Alcover_, Apr 14 2023 *)
%o A354157 (PARI) a(n) = numerator((1/3)*(1/(n+1/3))*prod(i=0, n-1, n+i+1/3)/n!) \\ _Rémy Sigrist_, May 30 2022
%Y A354157 Cf. A000108, A004128.
%K A354157 nonn,frac
%O A354157 0,3
%A A354157 _N. J. A. Sloane_, May 29 2022, based on Section 18.6 of Cosgrave (2022)
%E A354157 More terms from _Rémy Sigrist_, May 30 2022
%E A354157 a(0)=1 prepended by _Alois P. Heinz_, Apr 14 2023