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A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing. 3
1, 2, 9, 50, 295, 1792, 11088, 69498, 439791, 2803658, 17978389, 115837592, 749321716, 4863369656, 31655226108, 206549749930, 1350638103791, 8848643946550, 58069093513635, 381650672631330, 2511733593767295, 16550500379912640, 109176697072162080 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois 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]).

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

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Last modified April 26 17:21 EDT 2017. Contains 285449 sequences.