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A048742 a(n) = n! - (n-th Bell number).

%I A048742 #33 Aug 13 2020 22:31:27
%S A048742 0,0,0,1,9,68,517,4163,36180,341733,3512825,39238230,474788003,
%T A048742 6199376363,86987391878,1306291409455,20912309745853,355604563226196,
%U A048742 6401691628921841,121639267666626943,2432850284018404628,51090467301893283249,1123996221061869232677
%N A048742 a(n) = n! - (n-th Bell number).
%C A048742 Number of permutations of [n] which have at least one cycle that has at least one inversion when written with its smallest element in the first position. Example: a(4)=9 because we have (1)(243), (1432), (142)(3), (132)(4), (1342), (1423), (1243), (143)(2) and (1324). - _Emeric Deutsch_, Apr 29 2008
%C A048742 Number of permutations of [n] having consecutive runs of increasing elements with initial elements in increasing order. a(4) = 9: `124`3, `13`24, `134`2, `14`23, `14`3`2, `2`14`3, `24`3`1, `3`14`2, `4`13`2. - _Alois P. Heinz_, Apr 27 2016
%C A048742 From _Gus Wiseman_, Aug 11 2020: (Start)
%C A048742 Also the number of divisors of the superfactorial A006939(n - 1) without distinct prime multiplicities. For example, the a(4) = 9 divisors together with their prime signatures are the following. Note that A076954 can be used here instead of A006939.
%C A048742     6: (1,1)
%C A048742    10: (1,1)
%C A048742    15: (1,1)
%C A048742    30: (1,1,1)
%C A048742    36: (2,2)
%C A048742    60: (2,1,1)
%C A048742    90: (1,2,1)
%C A048742   120: (3,1,1)
%C A048742   180: (2,2,1)
%C A048742 (End)
%H A048742 Alois P. Heinz, <a href="/A048742/b048742.txt">Table of n, a(n) for n = 0..450</a>
%H A048742 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A048742 a(n) = A000142(n) - A000110(n).
%F A048742 E.g.f.: 1/(1-x) - exp(exp(x)-1). - _Alois P. Heinz_, Apr 27 2016
%p A048742 with(combinat): seq(factorial(n)-bell(n),n=0..21); # _Emeric Deutsch_, Apr 29 2008
%t A048742 Table[n! - BellB[n], {n, 0, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 06 2011 *)
%o A048742 (Sage) [factorial(m) - bell_number(m) for m in range(23)]  # _Zerinvary Lajos_, Jul 06 2008
%Y A048742 A000110 lists Bell numbers.
%Y A048742 A000142 lists factorial numbers.
%Y A048742 A006939 lists superprimorials or Chernoff numbers.
%Y A048742 A181796 counts divisors with distinct prime multiplicities.
%Y A048742 A336414 counts divisors of n! with distinct prime multiplicities.
%Y A048742 Cf. A000005, A022915, A027423, A076954, A095149, A130091, A336420, A336421.
%K A048742 nonn
%O A048742 0,5
%A A048742 _N. J. A. Sloane_