cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Peter Luschny, Aug 08 2010

Keywords

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

Crossrefs

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

Programs

  • 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 *)
  • 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

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

Extensions

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

A303285 Number of permutations p of [2n] such that the sequence of ascents and descents of p0 forms a Dyck path.

Original entry on oeis.org

1, 1, 8, 172, 7296, 518324, 55717312, 8460090160, 1726791794432, 456440969661508, 151770739970889792, 62022635037246022000, 30564038464166725328768, 17876875858414492985045712, 12245573879235563308351042496, 9711714975145772145881269175104
Offset: 0

Views

Author

Alois P. Heinz, Apr 20 2018

Keywords

Comments

Here p is a permutation of 1,2,3,...,2n, and p0 refers to the string p followed by 0.
Also the number of permutations p of [2n] such that the sequence of ascents and descents of 0p forms a Dyck path. a(2) = 8: 1432, 2143, 2431, 3142, 3241, 3421, 4132, 4231.
Also the number of permutations p of [2n] that are of odd order and whose M statistic (as defined in the Spiro paper) is equal to n-1. - Sam Spiro, Nov 01 2018

Examples

			a(0) = 1: the empty permutation.
a(1) = 1: 12.
a(2) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.

Links

Crossrefs

Bisection (even part) of A303284.
Bisection (even part) of A303287.
Column k=0 of A316728.
Cf. A177042, A180056, A316727, A321280.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(0, 2*n, 0):
seq(a(n), n=0..20);
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] + If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[0, 2n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 29 2018, from Maple *)
  • PARI
    \\ here b(n) is A177042
b(n)={if(n==0, 1, 2*sum(k=0, n, (-1)^k*binomial(2*n+1,k)*(n-k+1)^(2*n)));}
a(n)={if(n==0, 1, sum(k=1, n, binomial(2*n, 2*k-1)*b(k-1)*b(n-k))/2);} \\ Andrew Howroyd, Nov 01 2018

Formula

a(n) ~ c * 2^(2*n) * n^(2*n - 1) / exp(2*n), where c = 8.838022110416151362523442920999767406145711133564692... - Vaclav Kotesovec, May 22 2018
a(n) = (1/2)*Sum_{k odd} binomial(2*n,k)*A177042((k-1)/2)*A177042((2n-k-1)/2) for n>0. - Sam Spiro, Nov 01 2018
a(n) = A321280(2n,n-1) for n >= 1. - Alois P. Heinz, Nov 02 2018

A303287 Number of permutations p of [n] such that the sequence of ascents and descents of p or of p0 (if n is even) forms a Dyck path.

Original entry on oeis.org

1, 1, 1, 2, 8, 22, 172, 604, 7296, 31238, 518324, 2620708, 55717312, 325024572, 8460090160, 55942352184, 1726791794432, 12765597850950, 456440969661508, 3730771315561300, 151770739970889792, 1359124435588313876, 62022635037246022000, 603916464771468176392
Offset: 0

Views

Author

Alois P. Heinz, Apr 20 2018

Keywords

Examples

			a(0) = 1: the empty permutation.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 132, 231.
a(4) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
a(5) = 22: 12543, 13254, 13542, 14253, 14352, 14532, 15243, 15342, 23154, 23541, 24153, 24351, 24531, 25143, 25341, 34152, 34251, 34521, 35142, 35241, 45132, 45231.

Links

Crossrefs

Bisections give: A303285 (even part), A177042 (odd part).
Cf. A180056, A304001, A303284, A304777, A304778.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] + If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

Formula

a(2n) = A303284(2n).

A303286 Number of permutations p of [2n+1] such that the sequence of ascents and descents of 0p0 forms a Dyck path.

Original entry on oeis.org

1, 4, 60, 1974, 114972, 10490392, 1384890104, 250150900354, 59317740001132, 17886770092245360, 6687689652133397064, 3037468107154650475868, 1647659575564603380270360, 1052309674407466474533397824, 781725844087366504901991503920, 668408235613132734111402947167658
Offset: 0

Views

Author

Alois P. Heinz, Apr 20 2018

Keywords

Examples

			a(1) = 4: 132, 213, 231, 312.

Links

Crossrefs

Bisection (odd part) of A303284.
Cf. A180056.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
a:= n-> b(0, 2*n+1, 0):
seq(a(n), n=0..20);
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t > 0, Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] + If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
a[n_] := b[0, 2*n + 1, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 31 2018, from Maple *)

Formula

a(n) ~ c * 2^(2*n) * n^(2*n) / exp(2*n), where c = 45.0971960423271758887353825240016439879529954831112316... - Vaclav Kotesovec, May 22 2018
Showing 1-4 of 4 results.

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