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

%I A177042 #65 Jun 19 2024 08:09:41
%S A177042 1,2,22,604,31238,2620708,325024572,55942352184,12765597850950,
%T A177042 3730771315561300,1359124435588313876,603916464771468176392,
%U A177042 321511316149669476991132,202039976682357297272094824,147980747895225006590333244088,124963193751534047864734415925360
%N A177042 Eulerian version of the Catalan numbers, a(n) = A008292(2*n+1,n+1)/(n+1).
%C A177042 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
%C A177042 Twice the number of permutations of {1,2,...,2n} with n ascents. - _Peter Luschny_, Jan 11 2011
%H A177042 Alois P. Heinz, <a href="/A177042/b177042.txt">Table of n, a(n) for n = 0..220</a>
%H A177042 F. Ardila, <a href="http://dx.doi.org/10.1016/S0097-3165(03)00121-3">The Catalan matroid</a>, J. Combin. Theory Ser. A 104 (2003) 49-62.
%H A177042 Hoda Bidkhori and Seth Sullivant, <a href="http://arxiv.org/abs/1101.1108">Eulerian-Catalan Numbers</a>, arXiv:1101.1108 [math.CO], 2011.
%H A177042 Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/26.14#T1">Table 26.14.1</a>
%F A177042 a(n) = 2*A180056(n) for n > 0, A180056 the central Eulerian numbers in the sense of A173018.
%F A177042 a(n) = A025585(n+1)/(n+1), A025585 the central Eulerian numbers in the sense of A008292.
%F A177042 a(n) = 2 Sum_{k=0..n} (-1)^k binomial(2n+1,k) (n-k+1)^(2n).
%F A177042 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
%F A177042 a(n) = A008518(2n,n). - _Alois P. Heinz_, Jun 12 2017
%F A177042 From _Alois P. Heinz_, Jul 21 2018: (Start)
%F A177042 a(n) = (2n)! * [x^(2n) y^n] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x)).
%F A177042 a(n) = (2n+1)!/(n+1) * [x^(2n+1) y^(n+1)] (1-y)/(1-y*exp((1-y)*x)). (End)
%p A177042 A177042 := proc(n) A008292(2*n+1,n+1)/(n+1) ; end proc:
%p A177042 seq(A177042(n),n=0..10) ; # _R. J. Mathar_, Jan 08 2011
%p A177042 A177042 := n -> A025585(n+1)/(n+1):
%p A177042 A177042 := n -> `if`(n=0,1,2*A180056(n)):
%p A177042 # The A173018-based recursion below needs no division!
%p A177042 A := proc(n, k) option remember;
%p A177042        if n = 0 and k = 0 then 1
%p A177042      elif k > n  or k < 0 then 0
%p A177042      else (n-k) *A(n-1, k-1) +(k+1) *A(n-1, k)
%p A177042        fi
%p A177042      end:
%p A177042 A177042 := n-> `if`(n=0, 1, 2*A(2*n, n)):
%p A177042 seq(A177042(n), n=0..30);
%p A177042 # _Peter Luschny_, Jan 11 2011
%t A177042 << DiscreteMath`Combinatorica`
%t A177042 Table[(Eulerian[2*n + 1, n])/(n + 1), {n, 0, 20}]
%t A177042 (* Second program: *)
%t A177042 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_ *)
%o A177042 (Magma)
%o A177042 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]]) >;
%o A177042 [A177042(n): n in [0..40]]; // _G. C. Greubel_, Jun 18 2024
%o A177042 (SageMath)
%o A177042 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)
%o A177042 [A177042(n) for n in range(41)] # _G. C. Greubel_, Jun 18 2024
%Y A177042 Cf. A000108, A008518, A025585, A180056.
%Y A177042 Bisection (odd part) of A303287.
%Y A177042 Row sums of A316728.
%K A177042 nonn
%O A177042 0,2
%A A177042 _Roger L. Bagula_, May 01 2010
%E A177042 Edited by _Alois P. Heinz_, Jan 14 2011