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
Keywords
Examples
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)
Links
- G. C. Greubel and R. H. Hardin, Table of n, a(n) for n = 0..1000 (terms 1..51 from R. H. Hardin)
Programs
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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 *) -
PARI
{ a(n) = polcoeff( (1+x+O(x^(2*n+1)))^(-n-1)/(1-x), 2*n) }
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)
Extensions
a(0)=1 prepended by Alois P. Heinz, Feb 04 2017
Comments