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%I #44 Dec 18 2018 19:50:51
%S 1,1,11,302,15619,1310354,162512286,27971176092,6382798925475,
%T 1865385657780650,679562217794156938,301958232385734088196,
%U 160755658074834738495566,101019988341178648636047412,73990373947612503295166622044,62481596875767023932367207962680
%N The number of permutations of {1,2,...,2n} with n ascents.
%C 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 Then a(n) = A(2*n,n) are the central Eulerian numbers. (Analogous to what are called the central binomial coefficients).
%H Alois P. Heinz, <a href="/A180056/b180056.txt">Table of n, a(n) for n = 0..200</a>
%H Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/26.14#T1">Table 26.14.1</a>
%F a(n-1) = A025585(n)/(2*n). - _Gary Detlefs_, Nov 11 2011
%F a(n+1)/a(n) ~ 4*n^2. - _Ran Pan_, Oct 26 2015
%F a(n) ~ sqrt(3) * 2^(2*n+1) * n^(2*n) / exp(2*n). - _Vaclav Kotesovec_, Oct 16 2016
%F From _Alois P. Heinz_, Jul 21 2018: (Start)
%F a(n) = ceiling(1/2 * (2n)! * [x^(2n) y^n] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x))).
%F a(n) = (2n)! * [x^(2n) y^n] (1-y)/(1-y*exp((1-y)*x)). (End)
%p A180056 :=
%p proc(n) local j;
%p add((-1)^j*binomial(2*n+1,j)*(n-j+1)^(2*n),j=0..n)
%p end:
%p # A180056_list(m) returns [a_0,a_1,..,a_m]
%p A180056_list :=
%p proc(m) local A, R, M, n, k;
%p R := 1; M := m + 1;
%p A := array([seq(1, n = 1..M)]);
%p for n from 2 to M do
%p for k from 2 to M do
%p if n = k then R := R, A[k] fi;
%p A[k] := n*A[k-1] + k*A[k]
%p od
%p od;
%p R
%p end:
%t 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 << Combinatorica`; Table[Combinatorica`Eulerian[2 n, n], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 15 2016 *)
%o # Python
%o def A180056_list(m):
%o ....ret = [1]
%o ....M = m + 1
%o ....A = [1 for i in range(0, M)]
%o ....for n in range(2, M):
%o ........for k in range(2, M):
%o ............if n == k:
%o ................ret.append(A[k])
%o ............A[k] = n*A[k-1] + k*A[k]
%o ....return ret
%Y A bisection of A006551.
%Y Cf. A008292, A025585, A303284, A303285, A303286, A303287.
%Y A diagonal of A321967.
%K nonn
%O 0,3
%A _Peter Luschny_, Aug 08 2010
%E Partially edited by _N. J. A. Sloane_, Aug 08 2010
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