

A258399


Number of 4nlength 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
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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) * (2^31) = 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, ni)*(1)^i/((ni)!*i!), i=0..n):
seq(a(n), n=0..15);


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



