A064299 a(n) = B(n)*C(n), where B(n) are Bell numbers (A000110) and C(n) are Catalan numbers (A000108).

1, 1, 4, 25, 210, 2184, 26796, 376233, 5920200, 102816714, 1947916100, 39890416020, 876478739164, 20537052247300, 510548782729680, 13407568735200525, 370553407586717490, 10742998644116921160, 325786278993936753300, 10307990595756667951830

Offset: 0

Author

Karol A. Penson, Sep 05 2001

Keywords

• nonn

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

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.

Crossrefs

Cf. A000108, A000110.

Row sums of A253180.

Programs

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 *)

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))
A064299_list = list(islice(A064299_gen(),20)) # Chai Wah Wu, Jun 22 2022

Sage

[bell_number(i)*catalan_number(i) for i in range(17)] # Zerinvary Lajos, Mar 14 2009

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.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.