A387249 a(n) = 10/(n + 1) * Catalan(3*n).

10, 25, 440, 12155, 416024, 16158075, 682341000, 30582833775, 1433226830360, 69533550916004, 3468169547356640, 176946775343535925, 9199844912200348840, 486018122664268428850, 26029619941269629306160, 1410698658798280045783575, 77251704848334920869407000, 4269325372507953547350453420

Offset: 0

Author

Peter Bala, Aug 24 2025

Keywords

• nonn
• easy
• new

Comments

Compare with Catalan(n) = 1/(n + 1) * binomial(2*n, n).

For r >= 2, there is a constant K_r such that K_r/(n + 1) * Catalan(r*n) is integral for all n.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000108, A000984, A007272, A066802, A387248, A387250.

Programs

Maple

seq( 10/((n+1)*(3*n+1)) * binomial(6*n, 3*n), n = 0..20);

Mathematica

A387249[n_] := 10*CatalanNumber[3*n]/(n + 1); Array[A387249, 20, 0] (* Paolo Xausa, Sep 02 2025 *)

Formula

a(n) = 10/((n + 1)*(3*n + 1)) * binomial(6*n, 3*n).

a(n) = (3*n + 2)/2 * (16*Catalan(3*n) - 8*Catalan(3*n+1) + Catalan(3n+2)) (shows a(n) to be an integer since Catalan(n) is odd iff n = 2^k - 1 for some k, so Catalan(3*n+2) is always even).

a(n) = (3*n + 2)/2 * A007272(3*n).

a(n) = 8*(2*n - 1)*(6*n - 1)*(6*n - 5)/((n + 1)*(3*n + 1)*(3*n - 1)) * a(n-1) with a(0) = 10.

a(n) ~ 10/(sqrt(27*Pi)) * 64^n/n^(5/2).

E.g.f.: 10*hypergeom([1/6, 1/2, 5/6], [2/3, 4/3, 2], 64*x). - Stefano Spezia, Aug 27 2025