A258399 Number of 4n-length strings of balanced parentheses of exactly n different types that are introduced in ascending order.

1, 2, 98, 11880, 2432430, 714249900, 275335499824, 131928199603200, 75727786603836510, 50713478000403718500, 38843740303576863755100, 33508462196084294380001040, 32157574295254903735909896240, 33990046387543889224733323929120

Offset: 0

Author

Alois P. Heinz, May 28 2015

Keywords

• nonn

Examples

			a(0) = 1: the empty string.
a(1) = 2: ()(), (()).
a(2) = A000108(4) * (2^3-1) = 14*7 = 98.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A000108, A210029, A242446, A253180, A256254, A258426.

Programs

Maple

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
a:= n-> add(A(2*n, n-i)*(-1)^i/((n-i)!*i!), i=0..n):
seq(a(n), n=0..15);

Mathematica

A[n_, k_] := A[n, k] = k^n CatalanNumber[n];
a[n_] := If[n==0, 1, Sum[A[2n, n-i] (-1)^i/((n-i)! i!), {i, 0, n}]];
a /@ Range[0, 15] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *)

Formula

a(n) = A253180(2n,n).

a(n) ~ c * d^n * n! / n^(5/2), where d = A256254 = -64/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 98.8248737517356857317..., c = 1/(2^(5/2) * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.0412044746356859529237459292541572856326... . - Vaclav Kotesovec, Jun 01 2015, updated Sep 27 2023

a(n) = A210029(n) * (4*n)! / (n! * (2*n)! * (2*n + 1)!), for n>0. - Vaclav Kotesovec, Sep 27 2023