login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 17 19:40 EST 2017. Contains 294834 sequences.