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A020543 a(0) = 1, a(1) = 1, a(n+1) = (n+1)*a(n) + n. 8
1, 1, 3, 11, 47, 239, 1439, 10079, 80639, 725759, 7257599, 79833599, 958003199, 12454041599, 174356582399, 2615348735999, 41845579775999, 711374856191999, 12804747411455999, 243290200817663999, 4865804016353279999 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
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
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 *)
PROG
(Magma) [2*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013
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).
Sequence in context: A030865 A030902 A030925 * A111139 A167564 A295833
KEYWORD
nonn
AUTHOR
EXTENSIONS
Better description from Benoit Cloitre, Dec 29 2001
STATUS
approved

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Last modified April 17 15:33 EDT 2024. Contains 371764 sequences. (Running on oeis4.)