A088536 Number of unimodal functions [1..n]->[1..n].
1, 4, 22, 130, 791, 4900, 30738, 194634, 1241383, 7963384, 51325352, 332095816, 2155894508, 14035149748, 91593941402, 599021799242, 3924954250975, 25760310654100, 169322682857430, 1114452091832130, 7344021912458295, 48448974411575280, 319942093205166840, 2114743632331515480
Offset: 1
Keywords
Examples
From _Joerg Arndt_, May 10 2013: (Start) The a(3) = 22 unimodal maps [1,2,3]->[1,2,3] are 01: [ 1 1 1 ] 02: [ 1 1 2 ] 03: [ 1 1 3 ] 04: [ 1 2 1 ] 05: [ 1 2 2 ] 06: [ 1 2 3 ] 07: [ 1 3 1 ] 08: [ 1 3 2 ] 09: [ 1 3 3 ] 10: [ 2 1 1 ] 11: [ 2 2 1 ] 12: [ 2 2 2 ] 13: [ 2 2 3 ] 14: [ 2 3 1 ] 15: [ 2 3 2 ] 16: [ 2 3 3 ] 17: [ 3 1 1 ] 18: [ 3 2 1 ] 19: [ 3 2 2 ] 20: [ 3 3 1 ] 21: [ 3 3 2 ] 22: [ 3 3 3 ] (End)
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
Programs
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Mathematica
Table[Sum[Binomial[2k+n-1,2k],{k,0,n-1}],{n,1,20}] (* Vaclav Kotesovec, Oct 14 2012 *) -
PARI
a(n) = sum(k=0,n-1, binomial(2*k+n-1,2*k)); \\ Joerg Arndt, May 10 2013
Formula
a(n) = Sum_{k=0..n-1} binomial(2k+n-1,2k).
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
a(n) ~ 27^n/(5*2^(2*n-1)*sqrt(3*Pi*n)). - Vaclav Kotesovec, Oct 14 2012
It appears that a(n) = Sum_{k = 0..2*n-2} (-1)^k*binomial(n+k,k). - Peter Bala, Oct 08 2021
Extensions
More terms from David Wasserman, Aug 09 2005