A006551 -id:A006551 - cp's OEIS frontend

A154413 a(n) = A006551(n) - A018224(n).

0, 0, 0, 2, 30, 202, 2016, 14394, 151290, 1294478, 15660744, 162298842, 2274318228, 27968231436, 447527038848, 6382757516250, 114890215021650, 1865385066804550, 37307710791708600, 679562209260462054

Offset: 0

Roger L. Bagula, Jan 09 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A006551, A018224.

t[n_, k_] = Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
a0 = Table[t[n, Floor[n/2]], {n, 1, 30}];
b0 = Table[Binomial[n, Floor[(n)/2]]^2, {n, 0, 29}];
a=a0-b0

a(n) = Eulerian[n,Floor[n/2]] - Binomial[n,Floor[n/2]]^2.

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

Original entry on oeis.org

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

A154420 Maximal coefficient of MacMahon polynomial (cf. A060187) p(x,n)=2^n(1 - x)^(n + 1) LerchPhi[x, -n, 1/2]; that is, a(n) = Max(coefficients(p(x,n))).

Original entry on oeis.org

1, 1, 6, 23, 230, 1682, 23548, 259723, 4675014, 69413294, 1527092468, 28588019814, 743288515164, 16818059163492, 504541774904760, 13397724585164019, 455522635895576646, 13892023109165902550, 527896878148304296900

Offset: 0

Roger L. Bagula, Jan 09 2009

nonn

Since the center is the maximum in the Pascal, Eulerian and MacMahon triangles, a(n)=MacMahon[n,Floor[n/2]]

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Peter Luschny, Generalized Eulerian polynomials.

Cf. A060187, A006551.

gf := proc(n, k) local f; f := (x,t) -> x*exp(t*x/k)/(1-x*exp(t*x));
series(f(x,t), t, n+2); ((1-x)/x)^(n+1)*k^n*n!*coeff(%, t, n):
collect(simplify(%), x) end:
seq(coeff(gf(n,1),x,iquo(n,2)),n=0..18); # Middle Eulerian numbers, A006551.
seq(coeff(gf(n,2),x,iquo(n,2)),n=0..18); # Middle midpoint Eulerian numbers.
# Peter Luschny, May 02 2013

p[x_, n_] = 2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2];
Table[Max[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]], {n, 0, 30}]

a(n) ~ sqrt(3) * 2^(n+1) * n^n / exp(n). - Vaclav Kotesovec, Oct 28 2021

Edited by N. J. A. Sloane, Jan 15 2009

A304001 Number of permutations of [n] whose up-down signature has a nonnegative total sum.

Original entry on oeis.org

1, 1, 1, 5, 12, 93, 360, 3728, 20160, 259535, 1814400, 27820524, 239500800, 4251096402, 43589145600, 877606592736, 10461394944000, 235288904377275, 3201186852864000, 79476406782222500, 1216451004088320000, 33020655481590446318, 562000363888803840000

Offset: 0

Alois P. Heinz, May 04 2018

nonn

The up-down signature has (+1) for each ascent and (-1) for each descent.

Alois P. Heinz, Table of n, a(n) for n = 0..450

Bisections give: A002674 (even part), A179457(2n+1,n+1) (odd part).

Cf. A000246 (for nonnegative partial sums), A006551 (total sums are 0 or 1), A008292, A303287.

b:= proc(u, o, t) option remember; (n->
     `if`(t>=n, n!, `if`(t<-n, 0,
      add(b(u-j, o+j-1, t-1), j=1..u)+
      add(b(u+j-1, o-j, t+1), j=1..o))))(u+o)
    end:
a:= n-> `if`(n=0, 1, add(b(j-1, n-j, 0), j=1..n)):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> `if`(irem(n, 2, 'r')=0, ceil(n!/2),
         add(combinat[eulerian1](n, j), j=0..r)):
seq(a(n), n=0..25);

Eulerian1[n_, k_] := If[k == 0, 1, If[n == 0, 0, Sum[(-1)^j (k - j + 1)^n Binomial[n + 1, j], {j, 0, k + 1}]]];
a[n_] := Module[{r, m}, {r, m} = QuotientRemainder[n, 2]; If[m == 0, Ceiling[n!/2], Sum[Eulerian1[n, j], {j, 0, r}]]];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 26 2021, after 2nd Maple program *)

A154416 Maximal Stirling numbers of the first kind.

Original entry on oeis.org

1, 1, 1, 2, 11, 35, 274, 1624, 13068, 118124, 1026576, 12753576, 120543840, 1931559552, 20313753096, 392156797824, 5056995703824, 102992244837120, 1583313975727488, 34012249593822720, 610116075740491776, 13803759753640704000, 284093315901811468800

Offset: 0

Roger L. Bagula, Jan 09 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A002870, A006551, A065048.

Table[Max[Table[StirlingS1[n, m], {m, 0, n}]], {n, 0, 30}]

a(n) = vecmax(vector(n+1, m, stirling(n, m-1, 1))); \\ Michel Marcus, Sep 16 2016

a(n) = max_{m=0..n} StirlingS1(n,m).

cp's OEIS Frontend

A154413 a(n) = A006551(n) - A018224(n).

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

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A154420 Maximal coefficient of MacMahon polynomial (cf. A060187) p(x,n)=2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2]; that is, a(n) = Max(coefficients(p(x,n))).

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Maple

Mathematica

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Extensions

A304001 Number of permutations of [n] whose up-down signature has a nonnegative total sum.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A154416 Maximal Stirling numbers of the first kind.

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Crossrefs

Programs

Mathematica

PARI

Formula

A154420 Maximal coefficient of MacMahon polynomial (cf. A060187) p(x,n)=2^n(1 - x)^(n + 1) LerchPhi[x, -n, 1/2]; that is, a(n) = Max(coefficients(p(x,n))).