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

0, 0, 0, 1, 9, 68, 517, 4163, 36180, 341733, 3512825, 39238230, 474788003, 6199376363, 86987391878, 1306291409455, 20912309745853, 355604563226196, 6401691628921841, 121639267666626943, 2432850284018404628, 51090467301893283249, 1123996221061869232677

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

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

From Gus Wiseman, Aug 11 2020: (Start)

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.

6: (1,1)

10: (1,1)

15: (1,1)

30: (1,1,1)

36: (2,2)

60: (2,1,1)

90: (1,2,1)

120: (3,1,1)

180: (2,2,1)

(End)

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• Index entries for sequences related to factorial numbers

Crossrefs

A000110 lists Bell numbers.

A000142 lists factorial numbers.

A006939 lists superprimorials or Chernoff numbers.

A181796 counts divisors with distinct prime multiplicities.

A336414 counts divisors of n! with distinct prime multiplicities.

Cf. A000005, A022915, A027423, A076954, A095149, A130091, A336420, A336421.

Programs

Maple

with(combinat): seq(factorial(n)-bell(n),n=0..21); # Emeric Deutsch, Apr 29 2008

Mathematica

Table[n! - BellB[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)

Sage

[factorial(m) - bell_number(m) for m in range(23)]  # Zerinvary Lajos, Jul 06 2008

Formula

a(n) = A000142(n) - A000110(n).

E.g.f.: 1/(1-x) - exp(exp(x)-1). - Alois P. Heinz, Apr 27 2016