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A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

%I A386474 #14 Jul 24 2025 15:04:34
%S A386474 1,1,1,7,25,141,1171,9913,85233,907273,11010691,143824341,1988010553,
%T A386474 29605763773,475664908083,8284952367721,153508912353121,
%U A386474 2997209814190353,61485486404453443,1326994255131585373,30144049509450774441,718905298680190094341,17940822818538396541843
%N A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.
%C A386474 Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.
%H A386474 Alois P. Heinz, <a href="/A386474/b386474.txt">Table of n, a(n) for n = 0..300</a>
%F A386474 E.g.f.: Sum_{i>=0} ((x - x^(i+1))/(1 - x))^i / i!.
%e A386474 a(3) = 7 counts: {(1),(2),(3)}, {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}.
%p A386474 b:= proc(n, m, l) option remember; `if`(m>n+l, 0, `if`(n=0, 1,
%p A386474       add(b(n-j, max(m, j), l+1)*(n-1)!*j/(n-j)!, j=1..n)))
%p A386474     end:
%p A386474 a:= n-> b(n, 0$2):
%p A386474 seq(a(n), n=0..22);  # _Alois P. Heinz_, Jul 23 2025
%t A386474 With[{m = 22}, CoefficientList[1 + Series[Sum[((x - x^(i + 1))/(1 - x))^i/i!, {i, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* _Amiram Eldar_, Jul 24 2025 *)
%o A386474 (PARI) R_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(sum(i=0,N,((x-x^(i+1))/(1-x))^i/i!)))}
%Y A386474 Cf. A000262, A077229, A088009, A088311, A088312, A088313, A113775, A351884.
%K A386474 nonn,easy
%O A386474 0,4
%A A386474 _John Tyler Rascoe_, Jul 23 2025