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

0,2

Comments

a(0) = 1 by convention.

Links

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A074426 A082636 A242533 * A035603 A126735 A229679

Adjacent sequences: A273322 A273323 A273324 * A273326 A273327 A273328

Keyword

nonn,easy

Author

Alois P. Heinz, May 20 2016

Status

approved