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A088536 Number of unimodal functions [1..n]->[1..n].

%I A088536 #24 Oct 09 2021 06:53:56
%S A088536 1,4,22,130,791,4900,30738,194634,1241383,7963384,51325352,332095816,
%T A088536 2155894508,14035149748,91593941402,599021799242,3924954250975,
%U A088536 25760310654100,169322682857430,1114452091832130,7344021912458295,48448974411575280,319942093205166840,2114743632331515480
%N A088536 Number of unimodal functions [1..n]->[1..n].
%H A088536 Vincenzo Librandi, <a href="/A088536/b088536.txt">Table of n, a(n) for n = 1..200</a>
%F A088536 a(n) = Sum_{k=0..n-1} binomial(2k+n-1,2k).
%F A088536 Recurrence: 36*n*(2*n-3)*a(n) = 2*(269*n^2-549*n+235)*a(n-1) - (359*n^2-1062*n+907)*a(n-2) + 6*(3*n-8)*(3*n-7)*a(n-3). - _Vaclav Kotesovec_, Oct 14 2012
%F A088536 a(n) ~ 27^n/(5*2^(2*n-1)*sqrt(3*Pi*n)). - _Vaclav Kotesovec_, Oct 14 2012
%F A088536 It appears that a(n) = Sum_{k = 0..2*n-2} (-1)^k*binomial(n+k,k). - _Peter Bala_, Oct 08 2021
%e A088536 From _Joerg Arndt_, May 10 2013: (Start)
%e A088536 The a(3) = 22 unimodal maps [1,2,3]->[1,2,3] are
%e A088536 01:  [ 1 1 1 ]
%e A088536 02:  [ 1 1 2 ]
%e A088536 03:  [ 1 1 3 ]
%e A088536 04:  [ 1 2 1 ]
%e A088536 05:  [ 1 2 2 ]
%e A088536 06:  [ 1 2 3 ]
%e A088536 07:  [ 1 3 1 ]
%e A088536 08:  [ 1 3 2 ]
%e A088536 09:  [ 1 3 3 ]
%e A088536 10:  [ 2 1 1 ]
%e A088536 11:  [ 2 2 1 ]
%e A088536 12:  [ 2 2 2 ]
%e A088536 13:  [ 2 2 3 ]
%e A088536 14:  [ 2 3 1 ]
%e A088536 15:  [ 2 3 2 ]
%e A088536 16:  [ 2 3 3 ]
%e A088536 17:  [ 3 1 1 ]
%e A088536 18:  [ 3 2 1 ]
%e A088536 19:  [ 3 2 2 ]
%e A088536 20:  [ 3 3 1 ]
%e A088536 21:  [ 3 3 2 ]
%e A088536 22:  [ 3 3 3 ]
%e A088536 (End)
%t A088536 Table[Sum[Binomial[2k+n-1,2k],{k,0,n-1}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%o A088536 (PARI) a(n) = sum(k=0,n-1, binomial(2*k+n-1,2*k)); \\ _Joerg Arndt_, May 10 2013
%Y A088536 Main diagonal of A071920.
%Y A088536 Cf. A225006 (unimodal maps [1..n]->[1..n+1]).
%K A088536 nonn
%O A088536 1,2
%A A088536 Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 16 2003
%E A088536 More terms from _David Wasserman_, Aug 09 2005