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