A225006 -id:A225006 - cp's OEIS frontend

A088536 Number of unimodal functions [1..n]->[1..n].

1, 4, 22, 130, 791, 4900, 30738, 194634, 1241383, 7963384, 51325352, 332095816, 2155894508, 14035149748, 91593941402, 599021799242, 3924954250975, 25760310654100, 169322682857430, 1114452091832130, 7344021912458295, 48448974411575280, 319942093205166840, 2114743632331515480

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 16 2003

nonn

			From _Joerg Arndt_, May 10 2013: (Start)
The a(3) = 22 unimodal maps [1,2,3]->[1,2,3] are
01:  [ 1 1 1 ]
02:  [ 1 1 2 ]
03:  [ 1 1 3 ]
04:  [ 1 2 1 ]
05:  [ 1 2 2 ]
06:  [ 1 2 3 ]
07:  [ 1 3 1 ]
08:  [ 1 3 2 ]
09:  [ 1 3 3 ]
10:  [ 2 1 1 ]
11:  [ 2 2 1 ]
12:  [ 2 2 2 ]
13:  [ 2 2 3 ]
14:  [ 2 3 1 ]
15:  [ 2 3 2 ]
16:  [ 2 3 3 ]
17:  [ 3 1 1 ]
18:  [ 3 2 1 ]
19:  [ 3 2 2 ]
20:  [ 3 3 1 ]
21:  [ 3 3 2 ]
22:  [ 3 3 3 ]
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Main diagonal of A071920.

Cf. A225006 (unimodal maps [1..n]->[1..n+1]).

Table[Sum[Binomial[2k+n-1,2k],{k,0,n-1}],{n,1,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

a(n) = sum(k=0,n-1, binomial(2*k+n-1,2*k)); \\ Joerg Arndt, May 10 2013

a(n) = Sum_{k=0..n-1} binomial(2k+n-1,2k).
Recurrence: 36*n*(2*n-3)*a(n) = 2*(269*n^2-549*n+235)*a(n-1) - (359*n^2-1062*n+907)*a(n-2) + 6*(3*n-8)*(3*n-7)*a(n-3). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 27^n/(5*2^(2*n-1)*sqrt(3*Pi*n)). - Vaclav Kotesovec, Oct 14 2012
It appears that a(n) = Sum_{k = 0..2*n-2} (-1)^k*binomial(n+k,k). - Peter Bala, Oct 08 2021

More terms from David Wasserman, Aug 09 2005

A261668 Number of admissible words of Type G arising in study of q-analogs of multiple zeta values.

Original entry on oeis.org

1, 8, 49, 294, 1791, 11087, 69497, 439790, 2803657, 17978388, 115837591, 749321715, 4863369655, 31655226107, 206549749929, 1350638103790, 8848643946549, 58069093513634, 381650672631329, 2511733593767294, 16550500379912639, 109176697072162079, 720921085149563159

Offset: 1

N. J. A. Sloane, Sep 02 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..500
Mathoverflow, Asymptotics of A261668, 2015.
Jianqiang Zhao, Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, arXiv preprint arXiv:1412.8044 [math.NT], 2014. See Table 8, line 1.

b:= proc(n, t) option remember; `if`(t>n or t<0, 0,
      `if`(n=0, 1, add(j*b(n-j, t-1), j=1..n)))
    end:
a:= n-> add(add(b(d+k-1, d), d=1..n), k=1..n):
seq(a(n), n=1..25);  # Alois P. Heinz, Sep 06 2015

a[n_] := Sum[Binomial[2d+n-1, n-1], {d, 1, n}]; Array[a, 25] (* Jean-François Alcover, Feb 17 2016, after Max Alekseyev *)

a(n) = polcoeff(( (1+x+O(x^(2*n+1)))^(-n-1)-1)/(1-x), 2*n)

a(n) = A225006(n)-1.
a(n) = Sum_{1<=d,k<=n} Sum_{x1+···+xd=d+k-1 and x1,...,xd>=1} x1*x2*...*xd. See Proposition 10.8 p. 28 of Zhao link. - Michel Marcus, Sep 06 2015
a(n) = Sum_{d=1..n} binomial(2d+n-1,n-1). Also, a(n) is the coefficient of x^(2n) in ((1+x)^(-n-1)-1)/(1-x), or the coefficient of x^n in ((1+x)^(3n+1)-(1+x)^(n+1))/(2+x). - Max Alekseyev, Sep 14 2015

More terms from Alois P. Heinz, Sep 06 2015

A371815 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(4n-2k-1,n-2*k).

Original entry on oeis.org

1, 3, 20, 156, 1288, 10963, 95132, 836650, 7430956, 66501696, 598720080, 5416612336, 49201807276, 448442474938, 4099103160424, 37562606691526, 344959939645980, 3174051631201636, 29254814741949680, 270047153053464712, 2496167217049673468

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A225006, A262977, A371798.

Cf. A147855.

a(n) = sum(k=0, n\2, (-1)^k*binomial(4*n-2*k-1, n-2*k));

a(n) = [x^n] 1/((1+x^2) * (1-x)^(3*n)).

a(n) = binomial(4*n-1, n)*hypergeom([1, (1-n)/2, -n/2], [1/2-2*n, 1-2*n], -1). - Stefano Spezia, Apr 07 2024

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A088536 Number of unimodal functions [1..n]->[1..n].

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A261668 Number of admissible words of Type G arising in study of q-analogs of multiple zeta values.

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A371815 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(4n-2k-1,n-2*k).

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cp's OEIS Frontend

A088536 Number of unimodal functions [1..n]->[1..n].

Views

Author

Keywords

Examples

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Programs

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PARI

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Extensions

A261668 Number of admissible words of Type G arising in study of q-analogs of multiple zeta values.

Views

Author

Keywords

Links

Programs

Maple

Mathematica

PARI

Formula

Extensions

A371815 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(4*n-2*k-1,n-2*k).

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Author

Keywords

Crossrefs

Programs

PARI

Formula

A371815 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(4n-2k-1,n-2*k).