A025585 -id:A025585 - cp's OEIS frontend

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

1, 1, 11, 302, 15619, 1310354, 162512286, 27971176092, 6382798925475, 1865385657780650, 679562217794156938, 301958232385734088196, 160755658074834738495566, 101019988341178648636047412, 73990373947612503295166622044, 62481596875767023932367207962680

Offset: 0

Peter Luschny, Aug 08 2010

nonn

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).

Alois P. Heinz, Table of n, a(n) for n = 0..200
Digital Library of Mathematical Functions, Table 26.14.1

A bisection of A006551.
Cf. A008292, A025585, A123125, A303284, A303285, A303286, A303287.
A diagonal of A321967.

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:

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 *)

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

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

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

A177042 Eulerian version of the Catalan numbers, a(n) = A008292(2*n+1,n+1)/(n+1).

Original entry on oeis.org

1, 2, 22, 604, 31238, 2620708, 325024572, 55942352184, 12765597850950, 3730771315561300, 1359124435588313876, 603916464771468176392, 321511316149669476991132, 202039976682357297272094824, 147980747895225006590333244088, 124963193751534047864734415925360

Offset: 0

Roger L. Bagula, May 01 2010

nonn

According to the Bidkhori and Sullivant reference's abstract, authors show "that the Eulerian-Catalan numbers enumerate Dyck permutations, [providing] two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analog of the Chung-Feller theorem." - Jonathan Vos Post, Jan 07 2011

Twice the number of permutations of {1,2,...,2n} with n ascents. - Peter Luschny, Jan 11 2011

Alois P. Heinz, Table of n, a(n) for n = 0..220
F. Ardila, The Catalan matroid, J. Combin. Theory Ser. A 104 (2003) 49-62.
Hoda Bidkhori and Seth Sullivant, Eulerian-Catalan Numbers, arXiv:1101.1108 [math.CO], 2011.
Digital Library of Mathematical Functions, Table 26.14.1

Cf. A000108, A008518, A025585, A180056.
Bisection (odd part) of A303287.
Row sums of A316728.

A177042:=func< n | n eq 0 select 1 else 2*(&+[(-1)^k*Binomial(2*n+1,k)*(n-k+1)^(2*n): k in [0..n]]) >;
[A177042(n): n in [0..40]]; // G. C. Greubel, Jun 18 2024

A177042 := proc(n) A008292(2*n+1,n+1)/(n+1) ; end proc:
seq(A177042(n),n=0..10) ; # R. J. Mathar, Jan 08 2011
A177042 := n -> A025585(n+1)/(n+1):
A177042 := n -> `if`(n=0,1,2*A180056(n)):
# The A173018-based recursion below needs no division!
A := proc(n, k) option remember;
       if n = 0 and k = 0 then 1
     elif k > n  or k < 0 then 0
     else (n-k) *A(n-1, k-1) +(k+1) *A(n-1, k)
       fi
     end:
A177042 := n-> `if`(n=0, 1, 2*A(2*n, n)):
seq(A177042(n), n=0..30);
# Peter Luschny, Jan 11 2011

<< DiscreteMath`Combinatorica`
Table[(Eulerian[2*n + 1, n])/(n + 1), {n, 0, 20}]
(* Second program: *)
A[n_, k_] := A[n, k] = Which[n == 0 && k == 0, 1, k > n || k < 0, 0, True, (n - k)*A[n - 1, k - 1] + (k + 1)*A[n - 1, k]]; A177042[n_] := If[n == 0, 1, 2*A[2*n, n]]; Table[A177042[n], {n, 0, 30}] (* Jean-François Alcover, Jul 13 2017, after Peter Luschny *)

def A177042(n): return 2*sum((-1)^k*binomial(2*n+1,k)*(n-k+1)^(2*n) for k in range(n+1)) - int(n==0)
[A177042(n) for n in range(41)] # G. C. Greubel, Jun 18 2024

a(n) = 2*A180056(n) for n > 0, A180056 the central Eulerian numbers in the sense of A173018.
a(n) = A025585(n+1)/(n+1), A025585 the central Eulerian numbers in the sense of A008292.
a(n) = 2 Sum_{k=0..n} (-1)^k binomial(2n+1,k) (n-k+1)^(2n).
a(n) = (n+1)^(-1) Sum_{k=0..n} (-1)^k binomial(2n+2,k)(n+1-k)^(2n+1). - Peter Luschny, Jan 11 2011
a(n) = A008518(2n,n). - Alois P. Heinz, Jun 12 2017
From Alois P. Heinz, Jul 21 2018: (Start)
a(n) = (2n)! * [x^(2n) y^n] (exp(x)-y*exp(y*x))/(exp(y*x)-y*exp(x)).
a(n) = (2n+1)!/(n+1) * [x^(2n+1) y^(n+1)] (1-y)/(1-y*exp((1-y)*x)). (End)

Edited by Alois P. Heinz, Jan 14 2011

A006551 Maximal Eulerian numbers.

Original entry on oeis.org

1, 1, 4, 11, 66, 302, 2416, 15619, 156190, 1310354, 15724248, 162512286, 2275172004, 27971176092, 447538817472, 6382798925475, 114890380658550, 1865385657780650, 37307713155613000, 679562217794156938, 14950368791471452636

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

From Peter Luschny, Aug 08 2010: (Start)
Define A(n,k) as 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(n, floor(n/2)). The Digital Library of Mathematical Functions calls the A(n,k) Eulerian numbers. With this terminology a(n) are the middle Eulerian numbers and A180056 the central Eulerian numbers. (End)
Number of permutations of {1,2,..,n} with floor(n/2) descents. - Joerg Arndt, Aug 15 2014

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..450
Digital Library of Mathematical Functions, Table 26.14.1 [_Peter Luschny_, Aug 08 2010]
Herman Chau, On Enumerating Higher Bruhat Orders Through Deletion and Contraction, arXiv:2412.10532 [math.CO], 2024. See p. 20.
L. Lesieur and J.-N. Nicolas, On the Eulerian numbers M_n = max_{1<=k<=n} A(n,k), European J. Combin., 13 (1992), 379-399.
Robert G. Wilson v, Letter to N. J. A. Sloane, Apr. 1994

Cf. A008292. Bisections are A025585 and A180056.

a := proc(n) local j,k; k := iquo(n,2);
add((-1)^j*binomial(n+1,j)*(k-j+1)^n,j=0..k) end:
#  Peter Luschny, Aug 08 2010
# Computation by recursion:
A006551 := proc(r) local W; W := proc(m) local A,n,k;
A:=[seq(1, n=1..m)]; if m < 2 then RETURN(1) fi;
for n from 2 to m-1 do for k from 2 to m do
A[k] := n*A[k-1]+k*A[k] od od; [A[m-1],A[m]] end:
W((r+2+irem(r,2))/2)[2-irem(r,2)] end:
# Peter Luschny, Jan 12 2011

a[n_] := With[{k = Quotient[n, 2]}, Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k}]]; Array[a, 25] (* Jean-François Alcover, Feb 19 2017, after Peter Luschny *)

a(n) = sum_{0<=j<=floor(n/2)} (-1)^j binomial(n+1,j) (floor(n/2)-j+1)^n. [Peter Luschny, Aug 08 2010]
a(n+1)/a(n) ~ n. - Ran Pan, Oct 26 2015
a(n) ~ 2 * sqrt(3) * n^n / exp(n). - Vaclav Kotesovec, Oct 28 2021

A011818 Normalized volume of center slice of n-dimensional cube: 2^n* n!*Vol({ (x_1,...,x_n) in [ 0,1 ]^n: n/2 <= Sum_{i = 1..n} x_i <= (n+1)/2 }).

Original entry on oeis.org

1, 3, 16, 115, 1056, 11774, 154624, 2337507, 39984640, 763546234, 16101629952, 371644257582, 9319104528384, 252270887452380, 7332475985461248, 227761317947788323, 7529455986838732800, 263948439074152148450

Offset: 1

Guenter M. Ziegler (ziegler(AT)math.tu-berlin.de)

D. Chakerian, D. Logothetti, CubeSlices, Pictorial Triangles, and Probability, Math. Mag., Vol. 64 (1991) 219-241.

Cf. A008292, A025585, A104098.

a := n -> add(binomial(n,k)*eulerian1(n,k), k=0..n-1):
seq(a(n), n=1..18); # Peter Luschny, Jun 30 2016

Eulerian1[n_, k_] = Sum[(-1)^j (k-j+1)^n Binomial[n+1, j], {j, 0, k+1}];
a[n_] := Sum[Binomial[n, k] Eulerian1[n, k], {k, 0, n-1}];
Array[a, 18] (* Jean-François Alcover, Jun 03 2019 *)

V(d) = sum_{k=1}^{d-1} {d choose k-1} A_{d, k} where A_{k, d} denotes the Eulerian number (permutations of a d-set with k-1 descents) - see A008292.
Restated: a(n) = Sum_{k = 1..n} C(n,k-1)*A008292(n,k) for n>=1.
From Peter Bala, Jun 28 2016: (Start)
a(n) = 1/2*Sum_{k = 0..floor((n+1)/2)} (-1)^k*binomial(n + 1,k)*(n + 1 - 2*k)^n.
a(n) ~ sqrt(3)/2*(2/e)^(n+1)*(n+1)^n. (End)
a(2*n-1)/2^(2*n-2) = A025585(n) for n>=1. - Peter Luschny, Jun 30 2016

More terms from Paul D. Hanna, Mar 15 2006

A141693 Triangle read by rows: T(n,k) = (2k - n)A008292(n,k) with T(n,n) = n, 0 <= k <= n, where A008292 is the triangle of Eulerian numbers.

Original entry on oeis.org

0, -1, 1, -2, 0, 2, -3, -4, 1, 3, -4, -22, 0, 2, 4, -5, -78, -66, 26, 3, 5, -6, -228, -604, 0, 114, 4, 6, -7, -600, -3573, -2416, 1191, 360, 5, 7, -8, -1482, -17172, -31238, 0, 8586, 988, 6, 8, -9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9, -10

Offset: 0

Roger L. Bagula, Sep 09 2008

			Triangle begins:
    0;
   -1,     1;
   -2,     0,      2;
   -3,    -4,      1,       3;
   -4,   -22,      0,       2,       4;
   -5,   -78,    -66,      26,       3,     5;
   -6,  -228,   -604,       0,     114,     4,    6;
   -7,  -600,  -3573,   -2416,    1191,   360,    5,     7;
   -8, -1482, -17172,  -31238,       0,  8586,  988,     6, 8;
   -9, -3514, -73040, -264702, -156190, 88234, 43824, 2510, 7, 9;
  ...

Eric Weisstein's World of Mathematics, Eulerian Number
Wikipedia, Eulerian number

Cf. A008292.

T:= proc(n,k) `if`(n=k,n,(2*k-n)*add((-1)^j*(k-j+1)^n*binomial(n+1,j),j=0..k)); end proc: seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Oct 06 2018
T := (n, k) -> `if`(n = k, n, (2*k - n)*combinat:-eulerian1(n,k)):
seq(seq(T(n,k), k=0..n), n=0..9); # Peter Luschny, Oct 06 2018

T[n_, k_] = If[n == k, n, (2*k - n)*Sum[(-1)^j*(k - j + 1)^n*Binomial[n + 1, j], {j, 0, k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}]//Flatten

T(n, k) := if n = k then n else (2*k - n)*sum((-1)^j*(k - j + 1)^n*binomial(n + 1, j), j, 0, k)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 05 2018 */

Sum_{k=0..n} T(n,k) = A005096(n), n > 0.
From Franck Maminirina Ramaharo, Oct 06 2018: (Start)
T(n,k) = (2*k - n)*Sum_{j=0..k} (-1)^j*(k - j + 1)^n*binomial(n + 1, j) for 0 <= k <= n - 1 and T(n,n) = n.
T(2*n-1,n-1) = -A025585(n).
T(2*n,n-1) = -A177042(n). (End)

Edited, new name and offset corrected by Franck Maminirina Ramaharo, Oct 06 2018

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

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A177042 Eulerian version of the Catalan numbers, a(n) = A008292(2*n+1,n+1)/(n+1).

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A006551 Maximal Eulerian numbers.

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A011818 Normalized volume of center slice of n-dimensional cube: 2^n* n!*Vol({ (x_1,...,x_n) in [ 0,1 ]^n: n/2 <= Sum_{i = 1..n} x_i <= (n+1)/2 }).

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A141693 Triangle read by rows: T(n,k) = (2*k - n)*A008292(n,k) with T(n,n) = n, 0 <= k <= n, where A008292 is the triangle of Eulerian numbers.

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A141693 Triangle read by rows: T(n,k) = (2k - n)A008292(n,k) with T(n,n) = n, 0 <= k <= n, where A008292 is the triangle of Eulerian numbers.