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

1, 1, 21, 610, 25585, 1410003, 96451278, 7886294988, 750477171015, 81489964671115, 9943628857101511, 1347093605732587986, 200625344191782743506, 32581061387048389884550, 5729971899859216832319300, 1084929313931423899784882280, 220046254343980047765630634905

Offset: 0

Author

Alois P. Heinz, Sep 19 2023

Keywords

• nonn

Comments

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).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..313
• Wikipedia, Involution (mathematics)

Crossrefs

Cf. A001147, A079267.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 1, (8*(9*n-1)*(2*n-3)*a(n-2)+
      (n-1)*(243*n^3-513*n^2+360*n-76)*a(n-1))/(2*(9*n-10)*n*(n-1)))
    end:
seq(a(n), n=0..16);

Formula

a(n) = A079267(2n,n).