A387547 - OEIS

%I #17 Sep 08 2025 19:54:45

%S 1,0,1,1,36,211,6733,109110,3935179,120852049,5443425654,255558333097,

%T 14770120409245,943648512760656,69287778033581461,5659960611372895981,

%U 516263085381593167788,51978891572002399746295,5755459912529835391400569,696667046057392012794065454

%N Number of partitions of a 2n-set into even blocks of size > 2.

%C a(n) is the number of ways to form even size groups from a total of 2*n people, where each group has at least 4 people.

%F a(n) = (2*n)! * [x^(2*n)] exp(cosh(x) - x^2/2 - 1).

%F a(n) = Sum_{k=2..n} binomial(2*n-1, 2*k-1)*a(n-k) for n > 0, a(0)=1.

%e a(3) = 1 since for 6 people we can form a single group if a group has to have at least 4 people.

%e a(6) = 6733 since for 12 people the number of ways are (number of people in parentheses):

%e   1 group (12): 1 way;

%e   2 groups (8,4): 495 ways;

%e   2 groups (6,6): 462 ways;

%e   3 groups (4,4,4): 5775 ways.

%p b:= proc(n) option remember; `if`(n=0, 1,

%p       add(b(n-2*j)*binomial(n-1, 2*j-1), j=2..n/2))

%p     end:

%p a:= n-> b(2*n):

%p seq(a(n), n=0..19);  # _Alois P. Heinz_, Sep 01 2025

%t a[n_]:=(2*n)!*SeriesCoefficient[Exp[Cosh[x]-x^2/2-1],{x,0,2n}]; Array[a,20,0] (* _Stefano Spezia_, Sep 02 2025 *)

%Y Cf. A000110, A005046.

%K nonn,easy

%O 0,5

%A _Enrique Navarrete_, Sep 01 2025