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A064299 B(n)*C(n), where B(n) are Bell numbers (A000110) and C(n) are Catalan numbers (A000108). 4
1, 1, 4, 25, 210, 2184, 26796, 376233, 5920200, 102816714, 1947916100, 39890416020, 876478739164, 20537052247300, 510548782729680, 13407568735200525, 370553407586717490, 10742998644116921160, 325786278993936753300, 10307990595756667951830 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Feb 25 2017 *)

PROG

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

CROSSREFS

Cf. A000108, A000110.

Row sums of A253180.

Sequence in context: A105628 A332257 A203219 * A261898 A038174 A049118

Adjacent sequences:  A064296 A064297 A064298 * A064300 A064301 A064302

KEYWORD

nonn

AUTHOR

Karol A. Penson, Sep 05 2001

STATUS

approved

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Last modified January 26 07:39 EST 2021. Contains 340434 sequences. (Running on oeis4.)