cp's OEIS Frontend

A386497 Number of sets of lists of [n] such that one list is the largest.

%I A386497 #20 Aug 03 2025 03:27:25
%S A386497 1,1,2,12,60,440,3390,33852,338072,4116240,51776730,736751180,
%T A386497 11075784852,183142075272,3157190863190,59336602681020,
%U A386497 1164223828582320,24348331444705952,533422896546272562,12365952739192923660,298208300418298756460,7570420981014167756760
%N A386497 Number of sets of lists of [n] such that one list is the largest.
%C A386497 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 A386497 Alois P. Heinz, <a href="/A386497/b386497.txt">Table of n, a(n) for n = 0..444</a>
%F A386497 E.g.f.: 1 + Sum_{j>0} x^j * exp((x - x^j)/(1 - x)).
%e A386497 a(3) = 12 counts: {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}, {(1,2,3)}, {(1,3,2)}, {(2,1,3)}, {(2,3,1)}, {(3,1,2)}, {(3,2,1)}.
%p A386497 b:= proc(n, m, t) option remember; `if`(n=0, t, add(b(n-j, max(m, j),
%p A386497       `if`(j>m, 1, `if`(j=m, 0, t)))*(n-1)!*j/(n-j)!, j=1..n))
%p A386497     end:
%p A386497 a:= n-> b(n, 0, 1):
%p A386497 seq(a(n), n=0..21);  # _Alois P. Heinz_, Jul 23 2025
%t A386497 With[{m = 21}, CoefficientList[Series[1 + Sum[x^j*Exp[(x - x^j)/(1 - x)], {j, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* _Amiram Eldar_, Jul 24 2025 *)
%o A386497 (PARI) B_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(1+sum(j=1,N, x^j*exp((x-x^j)/(1-x)))))}
%Y A386497 Cf. A000262, A088009, A088311, A088312, A088313, A097979, A113775, A351884, A386474.
%K A386497 nonn,easy
%O A386497 0,3
%A A386497 _John Tyler Rascoe_, Jul 23 2025