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A258399 Number of 4n-length strings of balanced parentheses of exactly n different types that are introduced in ascending order. 4
1, 2, 98, 11880, 2432430, 714249900, 275335499824, 131928199603200, 75727786603836510, 50713478000403718500, 38843740303576863755100, 33508462196084294380001040, 32157574295254903735909896240, 33990046387543889224733323929120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

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

a(n) ~ c * d^n * n! / n^(5/2), where d = A256254 = 98.8248737517356857317..., c = 0.04120447463568595292374592925415728563265717672... . - Vaclav Kotesovec, Jun 01 2015

EXAMPLE

a(0) = 1: the empty string.

a(1) = 2: ()(), (()).

a(2) = A000108(4) * (2^3-1) = 14*7 = 98.

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-Fran├žois Alcover, Jan 01 2021, after Alois P. Heinz *)

CROSSREFS

Cf. A000108, A253180, A256254, A258426.

Sequence in context: A317729 A223038 A324266 * A212838 A024241 A278684

Adjacent sequences:  A258396 A258397 A258398 * A258400 A258401 A258402

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 28 2015

STATUS

approved

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Last modified May 14 22:40 EDT 2021. Contains 343909 sequences. (Running on oeis4.)