A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.

1, 2, 9, 50, 295, 1792, 11088, 69498, 439791, 2803658, 17978389, 115837592, 749321716, 4863369656, 31655226108, 206549749930, 1350638103791, 8848643946550, 58069093513635, 381650672631330, 2511733593767295, 16550500379912640, 109176697072162080

Offset

0,2

Comments

Diagonal of A225010.

Number of unimodal maps [1..n]->[1..n+1], see example. - Joerg Arndt, May 10 2013

Links

G. C. Greubel and R. H. Hardin, Table of n, a(n) for n = 0..1000 (terms 1..51 from R. H. Hardin)

Formula

From Vaclav Kotesovec, May 22 2013: (Start)

Empirical: 4*n*(2*n-1)*(5*n-7)*a(n) = 2*(145*n^3 - 343*n^2 + 235*n - 48)*a(n-1) - 3*(3*n-4)*(3*n-2)*(5*n-2)*a(n-2).

a(n) ~ 3^(3*n+3/2)/(5*2^(2*n+1)*sqrt(Pi*n)). (End)

a(n) = A261668(n)+1.

a(n) = Sum_{d=0..n} binomial(2d+n-1,n-1). Also, a(n) is the coefficient of x^(2n) in (1+x)^(-n-1)/(1-x). - Max Alekseyev, Sep 14 2015

It appears that a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k,k). - Peter Bala, Oct 08 2021

From Seiichi Manyama, Apr 06 2024: (Start)

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-2*k-1,n-2*k).

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

Example

Some solutions for n=3
..0..1..1....0..1..0....0..0..1....0..0..0....0..0..0....0..0..0....0..0..0
..1..1..1....0..1..0....1..1..1....0..0..0....0..0..0....0..1..0....0..0..1
..1..1..1....0..1..1....1..1..1....0..0..1....0..1..0....1..1..1....0..1..1
From Joerg Arndt, May 10 2013: (Start)
The a(2) = 9 unimodal maps [1,2]->[1,2,3] are
01:  [ 1 1 ]
02:  [ 1 2 ]
03:  [ 1 3 ]
04:  [ 2 1 ]
05:  [ 2 2 ]
06:  [ 2 3 ]
07:  [ 3 1 ]
08:  [ 3 2 ]
09:  [ 3 3 ]
(End)

Mathematica

a[n_] := Sum[Binomial[2d+n-1, n-1], {d, 0, n}]; Array[a, 30] (* Jean-François Alcover, Feb 17 2016, after Max Alekseyev *)

Prog

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

Crossrefs

Cf. A088536 (unimodal maps [1..n]->[1..n]).

Cf. A183160, A371798.

Sequence in context: A055997 A115599 A047069 * A211789 A192945 A271960

Adjacent sequences: A225003 A225004 A225005 * A225007 A225008 A225009

Keyword

nonn

Author

R. H. Hardin, Apr 23 2013

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 04 2017

Status

approved