A343665 - OEIS

%I #7 Apr 25 2021 20:07:38

%S 1,1,2,5,15,51,197,835,3860,19257,102997,586170,3535645,22496437,

%T 150454918,1054235150,7718958995,58905868192,467530598983,

%U 3851775136517,32881385742460,290387471713872,2649226725182823,24934118754400767,241809265181914545,2413608066257526577

%N Number of partitions of an n-set without blocks of size 5.

%F E.g.f.: exp(exp(x) - 1 - x^5/5!).

%F a(n) = n! * Sum_{k=0..floor(n/5)} (-1)^k * Bell(n-5*k) / ((n-5*k)! * k! * (5!)^k).

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

%p       `if`(j=5, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 25 2021

%t nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^5/5!], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[n! Sum[(-1)^k BellB[n - 5 k]/((n - 5 k)! k! (5!)^k), {k, 0, Floor[n/5]}], {n, 0, 25}]

%t a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 5, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

%Y Cf. A000110, A000296, A027339, A097514, A124504, A343664, A343666, A343667, A343668, A343669.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 25 2021