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

1, 1, 2, 5, 15, 51, 197, 835, 3860, 19257, 102997, 586170, 3535645, 22496437, 150454918, 1054235150, 7718958995, 58905868192, 467530598983, 3851775136517, 32881385742460, 290387471713872, 2649226725182823, 24934118754400767, 241809265181914545, 2413608066257526577

Offset: 0

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

• nonn

Crossrefs

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

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=5, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

Mathematica

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 5 k]/((n - 5 k)! k! (5!)^k), {k, 0, Floor[n/5]}], {n, 0, 25}]
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}]

Formula

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

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