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A180056 The number of permutations of {1,2,..,2n} with n ascents. 4
1, 1, 11, 302, 15619, 1310354, 162512286, 27971176092, 6382798925475, 1865385657780650, 679562217794156938, 301958232385734088196, 160755658074834738495566, 101019988341178648636047412, 73990373947612503295166622044, 62481596875767023932367207962680 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois 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

Cf. A008292, A025585. A bisection of A006551.

Sequence in context: A002114 A012192 A012079 * A172506 A250551 A001280

Adjacent sequences:  A180053 A180054 A180055 * A180057 A180058 A180059

KEYWORD

nonn

AUTHOR

Peter Luschny, Aug 08 2010

EXTENSIONS

Partially edited by N. J. A. Sloane, Aug 08 2010

STATUS

approved

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.