A343667 - OEIS

%I #7 Apr 25 2021 20:07:53

%S 1,1,2,5,15,52,203,876,4132,21075,115375,673620,4172413,27296089,

%T 187891174,1356343385,10238632307,80615222404,660560758879,

%U 5621465069117,49594663447612,452846969975391,4273130715906123,41612346388251187,417668648929556073,4315893703814296053

%N Number of partitions of an n-set without blocks of size 7.

%F E.g.f.: exp(exp(x) - 1 - x^7/7!).

%F a(n) = n! * Sum_{k=0..floor(n/7)} (-1)^k * Bell(n-7*k) / ((n-7*k)! * k! * (7!)^k).

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

%p       `if`(j=7, 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^7/7!], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[n! Sum[(-1)^k BellB[n - 7 k]/((n - 7 k)! k! (7!)^k), {k, 0, Floor[n/7]}], {n, 0, 25}]

%t a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 7, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

%Y Cf. A000110, A000296, A027341, A097514, A124504, A343664, A343665, A343666, A343668, A343669.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 25 2021