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

%I A273325 #26 Mar 12 2023 04:20:19
%S A273325 1,2,36,300,1960,11340,60984,312312,1544400,7438860,35103640,
%T A273325 162954792,746347056,3380195000,15164074800,67476121200,298135873440,
%U A273325 1309153089420,5717335239000,24847720451400,107520292479600,463440029892840,1990477619679120,8521600803066000
%N A273325 Number of endofunctions on [2n] such that the minimal cardinality of the nonempty preimages equals n.
%C A273325 a(0) = 1 by convention.
%H A273325 Alois P. Heinz, <a href="/A273325/b273325.txt">Table of n, a(n) for n = 0..1000</a>
%F A273325 G.f.: 1+(8*x+1)*2*x/(1-4*x)^(5/2).
%F A273325 a(n) = C(2*n,n)*C(2*n,2) for n>0, a(0)=1.
%F A273325 a(n) = 2*C(2*(n-1),n-1)*(2*n-1)^2, a(0)=1.
%F A273325 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.
%F A273325 a(n) = A245687(2n,n).
%F A273325 a(n) = A000108(n)*A213820(n) = 2*A000108(n)*A002414(n) for n>0, a(0)=1.
%F A273325 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
%e A273325 a(1) = 2: 12, 21.
%e A273325 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.
%p A273325 a:= proc(n) option remember; `if`(n<2, 2^n,
%p A273325        2*(2*n-1)^2*a(n-1)/((n-1)*(2*n-3)))
%p A273325     end:
%p A273325 seq(a(n), n=0..30);
%t A273325 a[n_] := (2*n^3 + n^2 - n) * CatalanNumber[n]; a[0] = 1; Array[a, 30, 0] (* _Amiram Eldar_, Mar 12 2023 *)
%Y A273325 Cf. A000108, A002414, A006752, A213820, A245687.
%K A273325 nonn,easy
%O A273325 0,2
%A A273325 _Alois P. Heinz_, May 20 2016