OFFSET
0,3
COMMENTS
From Joerg Arndt, Oct 22 2012: (Start)
Number of strings of length 2*n of up to n different types t(k) of balanced parentheses, where the first appearance of type t(k) must precede the appearance of t(k+1) for all k<n.
For example, from the 5 parenthesis string of length 3
1: ()()(); 2: ()(()); 3: (())(); 4: (()()); 5: ((())).
we obtain the B(3) * C(3) = 5 * 5 = 25 strings
1: ()()(), ()()[], ()[](), ()[][], ()[]{};
2: ()(()), ()([]), ()[()], ()[[]], ()[{}];
3: (())(), (())[], ([])(), ([])[], ([]){};
4: (()()), (()[]), ([]()), ([][]), ([]{});
5: ((())), (([])), ([()]), ([[]]), ([{}]).
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400
K. A. Penson and J.-M. Sixdeniers, Integral Representations of Catalan and Related Numbers, J. Integer Sequences, 4 (2001), #01.2.5.
FORMULA
Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n) = int(x^n*sum(sqrt((4*k-x)/x)*Heaviside(4*k-x)/(k*k!), k = 1..infinity)/(2*Pi*exp(1)), x = 0..infinity); this representation is unique.
MAPLE
with(combinat):
ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
a:= n-> bell(n)*ctln(n):
seq(a(n), n=0..25); # Alois P. Heinz, Mar 23 2015
MATHEMATICA
a[n_] := BellB[n]*CatalanNumber[n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 25 2017 *)
PROG
(Sage) [bell_number(i)*catalan_number(i) for i in range(17)] # Zerinvary Lajos, Mar 14 2009
(Python)
from itertools import count, accumulate, islice
def A064299_gen(): # generator of terms
yield from (1, 1)
blist, b, m = (1, 2), 1, 1
for n in count(1):
blist = list(accumulate(blist, initial=(b:=blist[-1])))
yield b*(m := m*(4*n+2)//(n+2))
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Sep 05 2001
STATUS
approved