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

%I A225006 #50 Apr 07 2024 11:24:33
%S A225006 1,2,9,50,295,1792,11088,69498,439791,2803658,17978389,115837592,
%T A225006 749321716,4863369656,31655226108,206549749930,1350638103791,
%U A225006 8848643946550,58069093513635,381650672631330,2511733593767295,16550500379912640,109176697072162080
%N A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.
%C A225006 Diagonal of A225010.
%C A225006 Number of unimodal maps [1..n]->[1..n+1], see example. - _Joerg Arndt_, May 10 2013
%H A225006 G. C. Greubel and R. H. Hardin, <a href="/A225006/b225006.txt">Table of n, a(n) for n = 0..1000</a> (terms 1..51 from R. H. Hardin)
%F A225006 From _Vaclav Kotesovec_, May 22 2013: (Start)
%F A225006 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).
%F A225006 a(n) ~ 3^(3*n+3/2)/(5*2^(2*n+1)*sqrt(Pi*n)). (End)
%F A225006 a(n) = A261668(n)+1.
%F A225006 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
%F A225006 It appears that a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k,k). - _Peter Bala_, Oct 08 2021
%F A225006 From _Seiichi Manyama_, Apr 06 2024: (Start)
%F A225006 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-2*k-1,n-2*k).
%F A225006 a(n) = [x^n] 1/((1+x^2) * (1-x)^(2*n)). (End)
%e A225006 Some solutions for n=3
%e A225006 ..0..1..1....0..1..0....0..0..1....0..0..0....0..0..0....0..0..0....0..0..0
%e A225006 ..1..1..1....0..1..0....1..1..1....0..0..0....0..0..0....0..1..0....0..0..1
%e A225006 ..1..1..1....0..1..1....1..1..1....0..0..1....0..1..0....1..1..1....0..1..1
%e A225006 From _Joerg Arndt_, May 10 2013: (Start)
%e A225006 The a(2) = 9 unimodal maps [1,2]->[1,2,3] are
%e A225006 01:  [ 1 1 ]
%e A225006 02:  [ 1 2 ]
%e A225006 03:  [ 1 3 ]
%e A225006 04:  [ 2 1 ]
%e A225006 05:  [ 2 2 ]
%e A225006 06:  [ 2 3 ]
%e A225006 07:  [ 3 1 ]
%e A225006 08:  [ 3 2 ]
%e A225006 09:  [ 3 3 ]
%e A225006 (End)
%t A225006 a[n_] := Sum[Binomial[2d+n-1, n-1], {d, 0, n}]; Array[a, 30] (* _Jean-François Alcover_, Feb 17 2016, after _Max Alekseyev_ *)
%o A225006 (PARI) { a(n) = polcoeff( (1+x+O(x^(2*n+1)))^(-n-1)/(1-x), 2*n) }
%Y A225006 Cf. A088536 (unimodal maps [1..n]->[1..n]).
%Y A225006 Cf. A183160, A371798.
%K A225006 nonn
%O A225006 0,2
%A A225006 _R. H. Hardin_, Apr 23 2013
%E A225006 a(0)=1 prepended by _Alois P. Heinz_, Feb 04 2017