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
KEYWORD
nonn
AUTHOR
R. H. Hardin, Apr 23 2013
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Feb 04 2017
STATUS
approved