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A180056 The number of permutations of {1,2,...,2n} with n ascents.

%I A180056 #50 Nov 13 2024 22:45:23
%S A180056 1,1,11,302,15619,1310354,162512286,27971176092,6382798925475,
%T A180056 1865385657780650,679562217794156938,301958232385734088196,
%U A180056 160755658074834738495566,101019988341178648636047412,73990373947612503295166622044,62481596875767023932367207962680
%N A180056 The number of permutations of {1,2,...,2n} with n ascents.
%C A180056 Define the Eulerian numbers A(n,k) (see A008292) to be the number of permutations of {1,2,..,n} with k ascents: A(n,k) = Sum_{j=0..k} (-1)^j binomial(n+1,j)*(k-j+1)^n.
%C A180056 Then a(n) = A(2*n,n) are the central Eulerian numbers. (Analogous to what are called the central binomial coefficients).
%H A180056 Alois P. Heinz, <a href="/A180056/b180056.txt">Table of n, a(n) for n = 0..200</a>
%H A180056 Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/26.14#T1">Table 26.14.1</a>
%F A180056 a(n-1) = A025585(n)/(2*n). - _Gary Detlefs_, Nov 11 2011
%F A180056 a(n+1)/a(n) ~ 4*n^2. - _Ran Pan_, Oct 26 2015
%F A180056 a(n) ~ sqrt(3) * 2^(2*n+1) * n^(2*n) / exp(2*n). - _Vaclav Kotesovec_, Oct 16 2016
%F A180056 From _Alois P. Heinz_, Jul 21 2018: (Start)
%F A180056 a(n) = ceiling(1/2 * (2n)! * [x^(2n) y^n] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x))).
%F A180056 a(n) = (2n)! * [x^(2n) y^n] (1-y)/(1-y*exp((1-y)*x)). (End)
%F A180056 a(n) = A123125(2n,n). - _Alois P. Heinz_, Nov 13 2024
%p A180056 A180056 :=
%p A180056 proc(n) local j;
%p A180056   add((-1)^j*binomial(2*n+1,j)*(n-j+1)^(2*n),j=0..n)
%p A180056 end:
%p A180056 # A180056_list(m) returns [a_0,a_1,..,a_m]
%p A180056 A180056_list :=
%p A180056   proc(m) local A, R, M, n, k;
%p A180056     R := 1; M := m + 1;
%p A180056     A := array([seq(1, n = 1..M)]);
%p A180056     for n from 2 to M do
%p A180056       for k from 2 to M do
%p A180056         if n = k then R := R, A[k] fi;
%p A180056         A[k] := n*A[k-1] + k*A[k]
%p A180056       od
%p A180056     od;
%p A180056   R
%p A180056 end:
%t A180056 A025585[n_] := Sum[(-1)^j*(n-j)^(2*n-1)*Binomial[2*n, j], {j, 0, n}]; a[0] = 1; a[n_] := A025585[n+1]/(2*n+2); Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Jun 28 2013, after _Gary Detlefs_ *)
%t A180056 << Combinatorica`; Table[Combinatorica`Eulerian[2 n, n], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 15 2016 *)
%o A180056 (Python)
%o A180056 def A180056_list(m):
%o A180056     ret = [1]
%o A180056     M = m + 1
%o A180056     A = [1 for i in range(0, M)]
%o A180056     for n in range(2, M):
%o A180056         for k in range(2, M):
%o A180056             if n == k:
%o A180056                 ret.append(A[k])
%o A180056             A[k] = n*A[k-1] + k*A[k]
%o A180056     return ret
%Y A180056 A bisection of A006551.
%Y A180056 Cf. A008292, A025585, A123125, A303284, A303285, A303286, A303287.
%Y A180056 A diagonal of A321967.
%K A180056 nonn
%O A180056 0,3
%A A180056 _Peter Luschny_, Aug 08 2010
%E A180056 Partially edited by _N. J. A. Sloane_, Aug 08 2010