A097514 - OEIS

%I #25 Jul 25 2023 09:32:46

%S 1,1,1,2,6,17,53,205,871,3876,18820,99585,558847,3313117,20825145,

%T 138046940,959298572,6974868139,52972352923,419104459913,

%U 3446343893607,29405917751526,259930518212766,2376498296500063,22441988298860757,218615700758838253

%N Number of partitions of an n-set without blocks of size 2.

%H Alois P. Heinz, <a href="/A097514/b097514.txt">Table of n, a(n) for n = 0..500</a>

%H Toufik Mansour and Mark Shattuck, <a href="https://doi.org/10.2298/AADM210223009M">Counting subword patterns in permutations arising as flattened partitions of sets</a>, Appl. Anal. Disc. Math. (2022), OnLine-First (00):9-9.

%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n, 2*k)*(2*k-1)!!*Bell(n-2*k).

%F E.g.f.: exp(exp(x)-1-x^2/2). More generally, e.g.f. for number of partitions of an n-set which contain exactly q blocks of size p is x^(p*q)/(q!*p!^q)*exp(exp(x)-1-x^p/p!).

%p g:=exp(exp(x)-1-x^2/2): gser:=series(g,x=0,31): 1,seq(n!*coeff(gser,x^n),n=1..29); # _Emeric Deutsch_, Nov 18 2004

%p # second Maple program:

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

%p        j=2, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 18 2015

%t a[n_] := a[n] = If[n == 0, 1, Sum[If[j == 2, 0, a[n-j]*Binomial[n-1, j-1]], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 13 2015, after _Alois P. Heinz_ *)

%Y Cf. A000296, A327885.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Aug 26 2004

%E More terms from _Emeric Deutsch_, Nov 18 2004