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

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

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

Example

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)

Mathematica

Table[Sum[Binomial[2k+n-1, 2k], {k, 0, n-1}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)

Prog

(PARI) a(n) = sum(k=0, n-1, binomial(2*k+n-1, 2*k)); \\ Joerg Arndt, May 10 2013

Crossrefs

Main diagonal of A071920.

Cf. A225006 (unimodal maps [1..n]->[1..n+1]).

Sequence in context: A086682 A261399 A155862 * A066380 A180899 A007195

Adjacent sequences: A088533 A088534 A088535 * A088537 A088538 A088539

Keyword

nonn

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 16 2003

Extensions

More terms from David Wasserman, Aug 09 2005

Status

approved