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

0,5

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

Formula

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

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

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 *)

Prog

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

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.

Sequence in context: A231192 A133120 A194650 * A121633 A091708 A327560

Adjacent sequences: A048739 A048740 A048741 * A048743 A048744 A048745

Keyword

nonn

Author

N. J. A. Sloane

Status

approved