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A006151
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Number of Dyck paths.
(Formerly M4288)
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4
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1, 1, 6, 91, 2548, 111384, 6852768, 553361016, 55804330152, 6774025632340, 962310111888300, 156490840602392625, 28622389306817092500, 5804104057179375825000, 1289547073500366035700000
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OFFSET
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0,3
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COMMENTS
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a(n) is the determinant of the 5 X 5 Hankel matrix [a_0, a_1, a_2, a_3, a_4 ; a_1, a_2, a_3, a_4, a_5 ; a_2, a_3, a_4, a_5, a_6 ; a_3, a_4, a_5, a_6, a_7 ; a_4, a_5, a_6, a_7, a_8] with a_j=A000108(n+j). - Philippe Deléham, Apr 12 2007
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REFERENCES
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M. de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire. Physique et Informatique. Ph.D Dissertation, Université Bordeaux I, 1983.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..200
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183).
M. de Sainte-Catherine and G. Viennot, Enumeration of certain Young tableaux with bounded height, in: G. Labelle and P. Leroux (eds), Combinatoire énumérative, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.
Nicholas M. Katz, A note on random matrix integrals, moment identities, and Catalan numbers, preprint, 2015.
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FORMULA
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From Vaclav Kotesovec, Mar 20 2014: (Start)
Recurrence: (n+5)*(n+6)*(n+7)*(n+8)*(n+9)*a(n) = 32*(2*n-1)*(2*n+1)*(2*n+3)*(2*n+5)*(2*n+7)*a(n-1).
a(n) = 1316818944000 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! * (2*(n+4))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)! * (n+8)! * (n+9)!).
a(n) ~ 1380784741023744000 * 1024^n / (Pi^(5/2) * n^(55/2)).
(End)
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MAPLE
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with(linalg): ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end: a:= n-> det(Matrix(5, (i, j)-> ctln(i+j-2+n))): seq(a(n), n=0..20); # Alois P. Heinz, Sep 10 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
32*mul((2*(n-i)+7)/(n+9-i), i=0..4)*a(n-1))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Mar 03 2014
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MATHEMATICA
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a[n_] := Det[Array[CatalanNumber[#1 + #2 - 2 + n]&, {5, 5}]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Table[1316818944000 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! * (2*(n+4))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)! * (n+8)! * (n+9)!), {n, 0, 20}] (* Vaclav Kotesovec, Mar 20 2014 *)
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CROSSREFS
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Sequence in context: A095864 A246155 A219220 * A005327 A182263 A171910
Adjacent sequences: A006148 A006149 A006150 * A006152 A006153 A006154
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Alois P. Heinz, Sep 10 2008
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STATUS
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approved
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