|
%I #40 Feb 05 2022 15:22:30
%S 1,1,3,7,16,33,71,143,295,594,1206,2413,4871,9743,19559,39138,78428,
%T 156857,314047,628095,1256809,2513693,5028594,10057189,20116979,
%U 40233975,80472823,160945945,321901713,643803427,1287627061,2575254123,5150547536,10301096282
%N a(1) = 1, a(n) = Sum_{k=1..n-1} Sum_{d|k} a(d).
%C Equals eigensequence of triangle A010766 and starting (1, 3, 7, 16, 33, ...) = row sums of triangle A163313. - _Gary W. Adamson_, Jul 30 2009. Gary Adamson's comment may be restated as "This sequence shifts left by one place under the floor transform." - _N. J. A. Sloane_, Feb 05 2016
%C The Gould & Quaintance reference, published in 2007, says incorrectly that this sequence is not in the OEIS. - _Olivier Gérard_, Oct 20 2011
%H Alois P. Heinz, <a href="/A014668/b014668.txt">Table of n, a(n) for n = 1..1000</a>
%H H. W. Gould and J. Quaintance, <a href="http://www.emis.de/journals/INTEGERS/papers/h58/h58.Abstract.html">Floor and Roof function analog of the Bell Numbers</a>, INTEGERS, 7 (2007), #A58.
%F a(n) is asymptotic to c*2^n where c = 0.59960731361450033896934...
%F a(n+1) = Sum_{k=1..n} a(k)*floor(n/k). - _Franklin T. Adams-Watters_, Mar 21 2017
%F G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * Sum_{k>=1} A(x^k)). - _Ilya Gutkovskiy_, Feb 25 2020
%p with(numtheory):
%p a:= proc(n) option remember;
%p `if`(n=1, 1, add(add(a(d), d=divisors(k)), k=1..n-1))
%p end:
%p seq(a(n), n=1..40); # _Alois P. Heinz_, Oct 28 2011
%t a[1] = 1; a[n_] := a[n] = Sum[Sum[a[d], {d, Divisors[k]}], {k, 1, n-1}]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Apr 07 2015 *)
%o (PARI) // an=vector(100); a(n)=if(n<0,0,an[n]); // an[1]=1; for(n=2,100,an[n]=sum(k=1,n-1,sumdiv(k,d,a(d))))
%Y Cf. A010766, A163313. - _Gary W. Adamson_, Jul 30 2009
%K nonn
%O 1,3
%A _Benoit Cloitre_, Jun 24 2003
|
