cp's OEIS Frontend

A365744 Number of perfect matchings on [2n] with n short pairs.

%I A365744 #27 Sep 19 2023 12:27:29
%S A365744 1,1,21,610,25585,1410003,96451278,7886294988,750477171015,
%T A365744 81489964671115,9943628857101511,1347093605732587986,
%U A365744 200625344191782743506,32581061387048389884550,5729971899859216832319300,1084929313931423899784882280,220046254343980047765630634905
%N A365744 Number of perfect matchings on [2n] with n short pairs.
%C A365744 Also number of fixed point free involutions on [4n] having n cycles with adjacent integers. a(2) = 21: (12)(34)(57)(68), (12)(35)(46)(78), (12)(35)(48)(67), (12)(37)(45)(68), (12)(37)(48)(56), (12)(38)(47)(56), (13)(24)(56)(78), (13)(26)(45)(78), (13)(28)(45)(67), (14)(23)(58)(67), (15)(23)(46)(78), (15)(23)(48)(67), (15)(26)(34)(78), (15)(28)(34)(67), (16)(25)(34)(78), (17)(23)(45)(68), (17)(23)(48)(56), (17)(28)(34)(56), (18)(23)(47)(56), (18)(25)(34)(67), (18)(27)(34)(56).
%H A365744 Alois P. Heinz, <a href="/A365744/b365744.txt">Table of n, a(n) for n = 0..313</a>
%H A365744 Wikipedia, <a href="https://en.wikipedia.org/wiki/Involution_(mathematics)">Involution (mathematics)</a>
%F A365744 a(n) = A079267(2n,n).
%p A365744 a:= proc(n) option remember; `if`(n<2, 1, (8*(9*n-1)*(2*n-3)*a(n-2)+
%p A365744       (n-1)*(243*n^3-513*n^2+360*n-76)*a(n-1))/(2*(9*n-10)*n*(n-1)))
%p A365744     end:
%p A365744 seq(a(n), n=0..16);
%Y A365744 Cf. A001147, A079267.
%K A365744 nonn
%O A365744 0,3
%A A365744 _Alois P. Heinz_, Sep 19 2023