A020543 a(0) = 1, a(1) = 1, a(n+1) = (n+1)*a(n) + n.

1, 1, 3, 11, 47, 239, 1439, 10079, 80639, 725759, 7257599, 79833599, 958003199, 12454041599, 174356582399, 2615348735999, 41845579775999, 711374856191999, 12804747411455999, 243290200817663999, 4865804016353279999

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

Comments

First Bernoulli polynomial evaluated at x=n! and multiplied by 2.

From Jaroslav Krizek, Jan 23 2010: (Start)

a(0) = 1, for n >= 1: a(n) = numbers m for which there is one iteration {floor(r/k)} for k = n, n-1, n-2, ... 1 with property r mod k = k-1 starting at r = m.

For n = 5: a(5) = 239;

floor(239/5) = 47, 239 mod 5 = 4;

floor( 47/4) = 11, 47 mod 4 = 3;

floor( 11/3) = 3, 11 mod 3 = 2;

floor( 3/2) = 1, 3 mod 2 = 1;

floor( 1/1) = 1, 1 mod 1 = 0. (End)

With offset 1, is the eigensequence of a triangle with the natural numbers (1, 2, 3, ...) as the right border, (1, 1, 2, 3, 4, ...) as the left border; and the rest zeros. - Gary W. Adamson, Aug 01 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Cf. A052898(n) - 2.

Cf. sequences of the type k*n!-1: A033312 (k=1), this sequence, A173323 (k=3), A173321 (k=4), A173317 (k=5), A173316 (k=6).

Programs

Magma

[2*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013

Mathematica

lst={1};s=1;Do[s+=(n+=s*n);AppendTo[lst, s], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)
FoldList[#1*#2 + #2 - 1 &, 1, Range[20]] (* Robert G. Wilson v, Jul 07 2012 *)
Table[2 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

Formula

E.g.f.: (-2 + exp(x) - x*exp(x))/(1-x). - Ralf Stephan, Feb 18 2004

a(n) = 2*n! - 1. - Gary W. Adamson, Jan 07 2008

a(0) = a(1) = 1, a(n) = a(n-1) * n + (n-1) for n >= 2. - Jaroslav Krizek, Jan 23 2010

a(n) ~ 2*sqrt(2*Pi*n)*n^n/exp(n). - Ilya Gutkovskiy, Aug 02 2016

Extensions

Better description from Benoit Cloitre, Dec 29 2001