A354157 Numerator of generalized Catalan number c_3(n) (see Comments).

1, 1, 5, 104, 836, 7315, 202895, 1949900, 19284511, 1754890501, 18058389349, 188502545504, 5973492827120, 63732573470888, 685813307216632, 22303841469480032, 243350841747362492, 2670252449037801100, 265034693078133749180, 2936064912067020698720

Offset: 0

Author

N. J. A. Sloane, May 29 2022, based on Section 18.6 of Cosgrave (2022)

Keywords

• nonn
• frac

Comments

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.

If 1/3 is everywhere changed to 1 we get the usual Catalan numbers A000108.

Examples

			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, ...
		

References

• J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.6.

Crossrefs

Cf. A000108, A004128.

Programs

Maple

c := proc(n) 1/3 * 1/(n+1/3) * mul(n + i + 1/3, i = 0..(n-1))/n!: end;

Mathematica

c3[n_] := With[{k = 3}, Pochhammer[n+1+1/k, n-1]/(k*n!)];
Table[Numerator[c3[n]], {n, 1, 19}] (* Jean-François Alcover, Apr 14 2023 *)

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

Extensions

More terms from Rémy Sigrist, May 30 2022

a(0)=1 prepended by Alois P. Heinz, Apr 14 2023