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

1, 1, 1, 7, 25, 141, 1171, 9913, 85233, 907273, 11010691, 143824341, 1988010553, 29605763773, 475664908083, 8284952367721, 153508912353121, 2997209814190353, 61485486404453443, 1326994255131585373, 30144049509450774441, 718905298680190094341, 17940822818538396541843

Offset

0,4

Comments

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Formula

E.g.f.: Sum_{i>=0} ((x - x^(i+1))/(1 - x))^i / i!.

Example

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)}.

Maple

b:= proc(n, m, l) option remember; `if`(m>n+l, 0, `if`(n=0, 1,
      add(b(n-j, max(m, j), l+1)*(n-1)!*j/(n-j)!, j=1..n)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 23 2025

Mathematica

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 *)

Prog

(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!)))}

Crossrefs

Cf. A000262, A077229, A088009, A088311, A088312, A088313, A113775, A351884.

Sequence in context: A082651 A241714 A151491 * A208425 A334651 A366941

Adjacent sequences: A386471 A386472 A386473 * A386475 A386476 A386477

Keyword

nonn,easy

Author

John Tyler Rascoe, Jul 23 2025

Status

approved