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A177042
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Eulerian version of the Catalan numbers, a(n) = A008292(2*n+1,n+1)/(n+1).
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11
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1, 2, 22, 604, 31238, 2620708, 325024572, 55942352184, 12765597850950, 3730771315561300, 1359124435588313876, 603916464771468176392, 321511316149669476991132, 202039976682357297272094824, 147980747895225006590333244088, 124963193751534047864734415925360
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OFFSET
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0,2
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COMMENTS
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According to the Bidkhori and Sullivant reference's abstract, authors show "that the Eulerian-Catalan numbers enumerate Dyck permutations, [providing] two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analog of the Chung-Feller theorem." - Jonathan Vos Post, Jan 07 2011
Twice the number of permutations of {1,2,...,2n} with n ascents. - Peter Luschny, Jan 11 2011
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LINKS
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FORMULA
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a(n) = 2 Sum_{k=0..n} (-1)^k binomial(2n+1,k) (n-k+1)^(2n).
a(n) = (n+1)^(-1) Sum_{k=0..n} (-1)^k binomial(2n+2,k)(n+1-k)^(2n+1). - Peter Luschny, Jan 11 2011
a(n) = (2n)! * [x^(2n) y^n] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x)).
a(n) = (2n+1)!/(n+1) * [x^(2n+1) y^(n+1)] (1-y)/(1-y*exp((1-y)*x)). (End)
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MAPLE
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# The A173018-based recursion below needs no division!
A := proc(n, k) option remember;
if n = 0 and k = 0 then 1
elif k > n or k < 0 then 0
else (n-k) *A(n-1, k-1) +(k+1) *A(n-1, k)
fi
end:
A177042 := n-> `if`(n=0, 1, 2*A(2*n, n)):
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MATHEMATICA
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<< DiscreteMath`Combinatorica`
Table[(Eulerian[2*n + 1, n])/(n + 1), {n, 0, 20}]
(* Second program: *)
A[n_, k_] := A[n, k] = Which[n == 0 && k == 0, 1, k > n || k < 0, 0, True, (n - k)*A[n - 1, k - 1] + (k + 1)*A[n - 1, k]]; A177042[n_] := If[n == 0, 1, 2*A[2*n, n]]; Table[A177042[n], {n, 0, 30}] (* Jean-François Alcover, Jul 13 2017, after Peter Luschny *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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