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Search: id:A131455
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| A131455 |
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Number of inequivalent properly oriented and labeled planar chord diagrams whose associated planar tree is a path on n+1 vertices. |
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4
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| 1, 2, 18, 284, 7280, 273246, 14144592, 965491288, 84027112704, 9081387766810, 1193283000239616, 187340544144604212, 34633340434838499328, 7446726867419368499894
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n) = n times the number of 2up-2down permutations of length 2n-1 = n*A005981(n-1) for n >= 2. a(n) ~ (c_1)*n*(2n-1)!/(c_2)^(2n), where c_1 is a constant and c_2 = 1.87510... is the smallest positive solution of the equation cos(z)* cosh(z)+1 = 0.
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LINKS
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B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J., 3 (2003), 647-659.
Eric Weisstein's World of Mathematics, Generalized Hyperbolic Functions.
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FORMULA
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E.g.f.: sum_{n = 1 .. inf} a(n)*(x^(2n))/(2n)! = (x/2)*(f(0,x)*f(1,x)- f(2,x)*f(3,x)+ f(3,x))/(f(0,x)^2 - f(1,x)*f(3,x)), where f(j,x) = sum_{k = 0 .. inf} (x^(4k+j))/(4k+j)!, j = 0,1,2,3, is the j th generalized hyperbolic function.
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CROSSREFS
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Cf. A005981, A131453, A131454.
Sequence in context: A032037 A138275 A127134 this_sequence A084947 A123385 A121564
Adjacent sequences: A131452 A131453 A131454 this_sequence A131456 A131457 A131458
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KEYWORD
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easy,nonn
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AUTHOR
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Peter Bala (pbala(AT)toucansurf.com), Jul 13 2007
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