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A006151
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Number of 5-tuples (p_1, p_2, ..., p_5) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}.
(Formerly M4288)
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7
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1, 1, 6, 91, 2548, 111384, 6852768, 553361016, 55804330152, 6774025632340, 962310111888300, 156490840602392625, 28622389306817092500, 5804104057179375825000, 1289547073500366035700000, 310827567433642575691950000, 80604345356574686019872460000
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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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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.
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FORMULA
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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)
a(n) = Product_{1 <= i <= j <= n-1} (i + j + 10)/(i + j).
a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 10)/(i + j - 1) for n >= 1. (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:
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MATHEMATICA
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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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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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