A273325 Number of endofunctions on [2n] such that the minimal cardinality of the nonempty preimages equals n.

1, 2, 36, 300, 1960, 11340, 60984, 312312, 1544400, 7438860, 35103640, 162954792, 746347056, 3380195000, 15164074800, 67476121200, 298135873440, 1309153089420, 5717335239000, 24847720451400, 107520292479600, 463440029892840, 1990477619679120, 8521600803066000

Offset: 0

Author

Alois P. Heinz, May 20 2016

Keywords

• nonn
• easy

Comments

a(0) = 1 by convention.

Examples

			a(1) = 2: 12, 21.
a(2) = 36: 1122, 1133, 1144, 1212, 1221, 1313, 1331, 1414, 1441, 2112, 2121, 2211, 2233, 2244, 2323, 2332, 2424, 2442, 3113, 3131, 3223, 3232, 3311, 3322, 3344, 3434, 3443, 4114, 4141, 4224, 4242, 4334, 4343, 4411, 4422, 4433.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A002414, A006752, A213820, A245687.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 2^n,
       2*(2*n-1)^2*a(n-1)/((n-1)*(2*n-3)))
    end:
seq(a(n), n=0..30);

Mathematica

a[n_] := (2*n^3 + n^2 - n) * CatalanNumber[n]; a[0] = 1; Array[a, 30, 0] (* Amiram Eldar, Mar 12 2023 *)

Formula

G.f.: 1+(8*x+1)*2*x/(1-4*x)^(5/2).

a(n) = C(2*n,n)*C(2*n,2) for n>0, a(0)=1.

a(n) = 2*C(2*(n-1),n-1)*(2*n-1)^2, a(0)=1.

a(n) = 2*(2*n-1)^2*a(n-1)/((n-1)*(2*n-3)) for n>1, a(n) = 2^n for n=0..1.

a(n) = A245687(2n,n).

a(n) = A000108(n)*A213820(n) = 2*A000108(n)*A002414(n) for n>0, a(0)=1.

Sum_{n>=0} 1/a(n) = 1 - log(sqrt(3)+2)*Pi/6 + 4*G/3, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 12 2023