A387250 a(n) = 105/(n + 1) * Catalan(4*n).

105, 735, 50050, 5460315, 742511070, 114872107350, 19348562209860, 3461691866723475, 647897423565562310, 125577883051534761666, 25029394494457424675100, 5103876046438721064520350, 1060725331955983336553011500, 224018752093294694626068131340, 47967198494914114482847609250184

Offset: 0

Author

Peter Bala, Aug 25 2025

Keywords

• nonn
• easy
• new

Comments

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

Crossrefs

Cf. A000108, A387248, A387249.

Programs

Maple

seq( 105/((n + 1)*(4*n + 1)) * binomial(8*n, 4*n), n = 0..20);

Mathematica

a[n_]:=105/(n+1)*CatalanNumber[4n];Array[a,15,0] (* James C. McMahon, Aug 29 2025 *)

Formula

a(n) = 105/((n + 1)*(4*n + 1)) * binomial(8*n, 4*n).

a(n) = 2*(8*n - 1)*(8*n - 3)*(8*n - 5)*(8*n - 7)/((n + 1)*(2*n - 1)*(4*n + 1)*(4*n - 1)) * a(n-1) with a(0) = 105.

a(n) ~ 105/(8*sqrt(Pi)) * 256^n/n^(5/2).

E.g.f.: 105*hypergeom([1/8, 3/8, 5/8, 7/8], [1/2, 3/4, 5/4, 2], 256*x). - Stefano Spezia, Aug 27 2025