OFFSET
0,3
COMMENTS
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.
Then a(n) = A(2*n,n) are the central Eulerian numbers. (Analogous to what are called the central binomial coefficients).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..200
Digital Library of Mathematical Functions, Table 26.14.1
FORMULA
a(n-1) = A025585(n)/(2*n). - Gary Detlefs, Nov 11 2011
a(n+1)/a(n) ~ 4*n^2. - Ran Pan, Oct 26 2015
a(n) ~ sqrt(3) * 2^(2*n+1) * n^(2*n) / exp(2*n). - Vaclav Kotesovec, Oct 16 2016
From Alois P. Heinz, Jul 21 2018: (Start)
a(n) = ceiling(1/2 * (2n)! * [x^(2n) y^n] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x))).
a(n) = (2n)! * [x^(2n) y^n] (1-y)/(1-y*exp((1-y)*x)). (End)
a(n) = A123125(2n,n). - Alois P. Heinz, Nov 13 2024
MAPLE
A180056 :=
proc(n) local j;
add((-1)^j*binomial(2*n+1, j)*(n-j+1)^(2*n), j=0..n)
end:
# A180056_list(m) returns [a_0, a_1, .., a_m]
A180056_list :=
proc(m) local A, R, M, n, k;
R := 1; M := m + 1;
A := array([seq(1, n = 1..M)]);
for n from 2 to M do
for k from 2 to M do
if n = k then R := R, A[k] fi;
A[k] := n*A[k-1] + k*A[k]
od
od;
R
end:
MATHEMATICA
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 *)
<< Combinatorica`; Table[Combinatorica`Eulerian[2 n, n], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2016 *)
PROG
(Python)
def A180056_list(m):
ret = [1]
M = m + 1
A = [1 for i in range(0, M)]
for n in range(2, M):
for k in range(2, M):
if n == k:
ret.append(A[k])
A[k] = n*A[k-1] + k*A[k]
return ret
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Aug 08 2010
EXTENSIONS
Partially edited by N. J. A. Sloane, Aug 08 2010
STATUS
approved