A258390 Number of 2n-length strings of balanced parentheses of exactly 2 different types that are introduced in ascending order.

2, 15, 98, 630, 4092, 27027, 181610, 1239810, 8582756, 60138078, 425800564, 3042175500, 21906338040, 158830645635, 1158564772890, 8496271312650, 62604582047700, 463275674416170, 3441483002640540, 25654715940496500, 191852749820189640, 1438895966711035950

Offset: 2

Author

Alois P. Heinz, May 28 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000

Crossrefs

Column k=2 of A253180.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, [0$2, 2][n+1],
      (2*n-1)*(6*n*a(n-1) -8*(2*n-3)*a(n-2))/(n*(n+1)))
    end:
seq(a(n), n=2..25);

Mathematica

Table[(2^(n-1)-1)*Binomial[2n,n]/(n+1),{n,2,20}] (* Vaclav Kotesovec, Jun 01 2015 *)

Formula

a(n) = (2*n-1)*(6*n*a(n-1)-8*(2*n-3)*a(n-2))/(n*(n+1)) for n>2, a(2)=2, a(n)=0 for n<2.

a(n) = (2^(n-1)-1) * binomial(2n,n)/(n+1) = (2^(n-1)-1)*A000108(n). - Vaclav Kotesovec, Jun 01 2015