A094258 a(1) = 1, a(n+1) = n*n! for n >= 1.

1, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000, 48658040163532800000, 1072909785605898240000, 24728016011107368960000, 594596384994354462720000

Offset

1,3

Comments

The old definition was: "a(1) = 1, a(n+1) = n*(a(1) + a(2) + ... + a(n))."

a(n) is the number of positive integers k <= n! such that k is not divisible by n. It is also the number of rationals in (0,1] which can be written in the form m/n! but not in the form m/(n-1)!. - Jonathan Sondow, Aug 14 2006

Also, the number of monomials in the determinant of an n X n symbolic matrix with only one zero entry. The position of the zero in the matrix is not important. - Artur Jasinski, Apr 02 2008

From Zak Seidov, Jun 21 2005: (Start)

The number of integers that use each of the decimal digits 0 through n exactly once is the finite sequence 1, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, because there are (n+1)! permutations of decimal digits 0 through n, from which we remove the n! permutations with leading zero and get n*n! = total number of integers that use each of the decimal digits from 0 through n exactly once. For n=0 we have 1 integer (0) which uses zero exactly once, hence a(0)=1 by definition.

This sequence is finite because there are only 10 decimal digits. With the initial 1 replaced by 0, we get the initial terms of A001563, which is infinite. Cf. A109075 = number of primes which use each of the decimal digits from 0 through n exactly once. (End)

Partial sums yield factorial numbers A000142(n) = n! = (1, 2, 6, 24, 120, 720, ...). - Vladimir Joseph Stephan Orlovsky, Jun 27 2009

Links

Harvey P. Dale, Table of n, a(n) for n = 1..400

Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.

Formula

a(n+1) = n*n! = A001563(n) for n>=1.

From Jonathan Sondow, Aug 14 2006: (Start)

a(n) = n! - (n-1)! for n >= 2.

a(n) = n! - a(n-1) - a(n-2) - ... - a(1). with a(1) = 1. (End)

a(n) = A094304(n+1) = A001563(n-1) for n >= 2. - Jaroslav Krizek, Oct 16 2009

G.f.: 1/Q(0), where Q(k)= 1 + x/(1-x) - x/(1-x)*(k+2)/(1 - x/(1-x)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013

G.f.: W(0)*(1-sqrt(x)), where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+2)/(sqrt(x)*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 18 2013

Example

a(1) = 1;
a(2) = 1*a(1) = 1;
...
a(7) = 6*(a(1) + a(2) + ... + a(6)) = 6*(1 + 1 + 4 + 18 + 96 + 600) = 4320.

Maple

A094258 := proc(n) option remember: if n = 1 then 1; else (n-1)*add(A094258(i), i=1..n-1) ; fi ; end: seq(A094258(n), n=1..24) ; # R. J. Mathar, Jul 27 2007

Mathematica

a=s=1; lst={a}; Do[a=s*n-s; s+=a; AppendTo[lst, a], {n, 2, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 27 2009 *)
Module[{lst={1}}, Do[AppendTo[lst, n*Total[lst]], {n, 30}]; lst] (* Harvey P. Dale, Jul 01 2012 *)

Prog

(PARI) A094258(n)=(n-1)!*(n-(n>1)) \\ M. F. Hasler, Oct 21 2012

Crossrefs

Up to the offset and initial value, the same as A001563, cf. formula.

Cf. A109075.

Sequence in context: A152392 A001563 A094304 * A234854 A334735 A086681

Adjacent sequences: A094255 A094256 A094257 * A094259 A094260 A094261

Keyword

nonn

Author

Amarnath Murthy, Apr 26 2004

Extensions

Edited by Mark Hudson, Jan 05 2005

More terms from R. J. Mathar, Jul 27 2007

Edited by M. F. Hasler, Oct 21 2012

Edited by Jon E. Schoenfield, Jan 17 2015

Definition simplified by M. F. Hasler, Jun 28 2016

Status

approved