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A064299
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B(n)*C(n), where B(n) are Bell numbers (A000110) and C(n) are Catalan numbers (A000108).
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4
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1, 1, 4, 25, 210, 2184, 26796, 376233, 5920200, 102816714, 1947916100, 39890416020, 876478739164, 20537052247300, 510548782729680, 13407568735200525, 370553407586717490, 10742998644116921160, 325786278993936753300, 10307990595756667951830
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OFFSET
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0,3
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COMMENTS
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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)
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LINKS
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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.
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FORMULA
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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.
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MAPLE
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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
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MATHEMATICA
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a[n_] := BellB[n]*CatalanNumber[n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 25 2017 *)
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PROG
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(Sage) [bell_number(i)*catalan_number(i) for i in range(17)] # Zerinvary Lajos, Mar 14 2009
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CROSSREFS
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Cf. A000108, A000110.
Row sums of A253180.
Sequence in context: A105628 A332257 A203219 * A261898 A038174 A049118
Adjacent sequences: A064296 A064297 A064298 * A064300 A064301 A064302
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KEYWORD
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nonn
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AUTHOR
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Karol A. Penson, Sep 05 2001
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STATUS
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approved
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