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A006150
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Number of 4-tuples (p_1, p_2, ..., p_4) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}.
(Formerly M4013)
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8
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1, 1, 5, 55, 1001, 26026, 884884, 37119160, 1844536720, 105408179176, 6774025632340, 481155055944150, 37259723952950625, 3111129272480118750, 277587585343361452500, 26268551497229678505000, 2620002484114994890890000, 273961129317241857069150000, 29896847445736985488399170000
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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 4 X 4 Hankel matrix [a_0, a_1, a_2, a_3 ; a_1, a_2, a_3, a_4 ; a_2, a_3, a_4, a_5 ; a_3, a_4, a_5, a_6] with a_j=A000108(n+j). - Philippe Deléham, Apr 12 2007
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REFERENCES
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S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183).
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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FORMULA
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a(n) = Det[Table[binomial[i+3, j-i+4], {i, 1, n}, {j, 1, n}]]. - David Callan, Jul 20 2005
Recurrence: (n+4)*(n+5)*(n+6)*(n+7)*a(n) = 16*(2*n-1)*(2*n+1)*(2*n+3)*(2*n+5)*a(n-1).
a(n) = 3628800 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)!).
a(n) ~ 14863564800 * 256^n / (Pi^2 * n^18). (End)
a(n) = Product_{1 <= i <= j <= n-1} (i + j + 8)/(i + j).
a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 8)/(i + j - 1) for n >= 1. (End)
E.g.f.: hypergeom([1/2, 3/2, 5/2, 7/2], [5, 6, 7, 8], 256*x). - Stefano Spezia, Dec 09 2023
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MAPLE
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with(LinearAlgebra):
ctln:= proc(n) option remember; binomial(2*n, n)/ (n+1) end:
a:= n-> Determinant(Matrix(4, (i, j)-> ctln(i+j-2+n))):
seq(a(n), n=0..20); # Alois P. Heinz, Sep 10 2008, revised, Sep 05 2019
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MATHEMATICA
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Join[{1}, Table[Det[Table[Binomial[i+3, j-i+4], {i, n}, {j, n}]], {n, 20}]] (* Harvey P. Dale, Jul 31 2012 *)
Table[3628800 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)!), {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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