A177042 Eulerian version of the Catalan numbers, a(n) = A008292(2*n+1,n+1)/(n+1).

1, 2, 22, 604, 31238, 2620708, 325024572, 55942352184, 12765597850950, 3730771315561300, 1359124435588313876, 603916464771468176392, 321511316149669476991132, 202039976682357297272094824, 147980747895225006590333244088, 124963193751534047864734415925360

Offset: 0

Author

Roger L. Bagula, May 01 2010

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..220
• F. Ardila, The Catalan matroid, J. Combin. Theory Ser. A 104 (2003) 49-62.
• Hoda Bidkhori and Seth Sullivant, Eulerian-Catalan Numbers, arXiv:1101.1108 [math.CO], 2011.
• Digital Library of Mathematical Functions, Table 26.14.1

Crossrefs

Cf. A000108, A008518, A025585, A180056.

Bisection (odd part) of A303287.

Row sums of A316728.

Programs

Magma

A177042:=func< n | n eq 0 select 1 else 2*(&+[(-1)^k*Binomial(2*n+1,k)*(n-k+1)^(2*n): k in [0..n]]) >;
[A177042(n): n in [0..40]]; // G. C. Greubel, Jun 18 2024

Maple

A177042 := proc(n) A008292(2*n+1,n+1)/(n+1) ; end proc:
seq(A177042(n),n=0..10) ; # R. J. Mathar, Jan 08 2011
A177042 := n -> A025585(n+1)/(n+1):
A177042 := n -> `if`(n=0,1,2*A180056(n)):
# 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)):
seq(A177042(n), n=0..30);
# Peter Luschny, Jan 11 2011

Mathematica

<< 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 *)

SageMath

def A177042(n): return 2*sum((-1)^k*binomial(2*n+1,k)*(n-k+1)^(2*n) for k in range(n+1)) - int(n==0)
[A177042(n) for n in range(41)] # G. C. Greubel, Jun 18 2024

Formula

a(n) = 2*A180056(n) for n > 0, A180056 the central Eulerian numbers in the sense of A173018.

a(n) = A025585(n+1)/(n+1), A025585 the central Eulerian numbers in the sense of A008292.

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) = A008518(2n,n). - Alois P. Heinz, Jun 12 2017

From Alois P. Heinz, Jul 21 2018: (Start)

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)

Extensions

Edited by Alois P. Heinz, Jan 14 2011