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A343319 Number of ways to partition n labeled elements into sets of different sizes of at least 4.

%I A343319 #16 Apr 28 2021 20:49:44
%S A343319 1,0,0,0,1,1,1,1,1,127,211,793,1288,3719,6007,646439,1467077,7211843,
%T A343319 30123763,91160937,293184840,1118980377,110635063749,319072758997,
%U A343319 1918239941962,9518126978941,58119248603131,202992067559011,1031021295578251,4151156602678042,650225250329137612
%N A343319 Number of ways to partition n labeled elements into sets of different sizes of at least 4.
%H A343319 Alois P. Heinz, <a href="/A343319/b343319.txt">Table of n, a(n) for n = 0..700</a>
%F A343319 E.g.f.: Product_{k>=4} (1 + x^k/k!).
%p A343319 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A343319      `if`(i>n, 0, b(n, i+1)+b(n-i, i+1)*binomial(n, i)))
%p A343319     end:
%p A343319 a:= n-> b(n, 4):
%p A343319 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 28 2021
%t A343319 nmax = 30; CoefficientList[Series[Product[(1 + x^k/k!), {k, 4, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A343319 a[0] = 1; a[n_] := a[n] = -(n - 1)! Sum[DivisorSum[k, # (-#!)^(-k/#) &, # > 3 &] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 30}]
%Y A343319 Cf. A007837, A025149, A032311, A057837, A232475, A341283, A343542.
%K A343319 nonn
%O A343319 0,10
%A A343319 _Ilya Gutkovskiy_, Apr 28 2021