cp's OEIS Frontend

A343665 Number of partitions of an n-set without blocks of size 5.

%I A343665 #7 Apr 25 2021 20:07:38
%S A343665 1,1,2,5,15,51,197,835,3860,19257,102997,586170,3535645,22496437,
%T A343665 150454918,1054235150,7718958995,58905868192,467530598983,
%U A343665 3851775136517,32881385742460,290387471713872,2649226725182823,24934118754400767,241809265181914545,2413608066257526577
%N A343665 Number of partitions of an n-set without blocks of size 5.
%F A343665 E.g.f.: exp(exp(x) - 1 - x^5/5!).
%F A343665 a(n) = n! * Sum_{k=0..floor(n/5)} (-1)^k * Bell(n-5*k) / ((n-5*k)! * k! * (5!)^k).
%p A343665 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A343665       `if`(j=5, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
%p A343665     end:
%p A343665 seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 25 2021
%t A343665 nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
%t A343665 Table[n! Sum[(-1)^k BellB[n - 5 k]/((n - 5 k)! k! (5!)^k), {k, 0, Floor[n/5]}], {n, 0, 25}]
%t A343665 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 A343665 Cf. A000110, A000296, A027339, A097514, A124504, A343664, A343666, A343667, A343668, A343669.
%K A343665 nonn
%O A343665 0,3
%A A343665 _Ilya Gutkovskiy_, Apr 25 2021