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Number of partitions of an n-set without blocks of size 4.
8

%I #10 Apr 28 2021 20:59:44

%S 1,1,2,5,14,47,173,702,3125,14910,76495,418035,2418397,14791597,

%T 95093612,641094695,4521228732,33250447919,254585084539,2024995604762,

%U 16702070759557,142642458681486,1259387604241013,11479967000116911,107910143688962037,1044735841257587203,10407104137208385924

%N Number of partitions of an n-set without blocks of size 4.

%H Alois P. Heinz, <a href="/A343664/b343664.txt">Table of n, a(n) for n = 0..581</a>

%F E.g.f.: exp(exp(x) - 1 - x^4/4!).

%F a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * Bell(n-4*k) / ((n-4*k)! * k! * (4!)^k).

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

%p `if`(j=4, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))

%p end:

%p seq(a(n), n=0..26); # _Alois P. Heinz_, Apr 25 2021

%t nmax = 26; CoefficientList[Series[Exp[Exp[x] - 1 - x^4/4!], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[n! Sum[(-1)^k BellB[n - 4 k]/((n - 4 k)! k! (4!)^k), {k, 0, Floor[n/4]}], {n, 0, 26}]

%t a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 4, 0, Binomial[n - 1, k - 1] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

%Y Cf. A000110, A000296, A027338, A097514, A124504, A343665, A343666, A343667, A343668, A343669.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 25 2021