A097939 - OEIS

%I #51 Oct 15 2024 17:13:31

%S 1,3,6,12,22,42,79,151,291,566,1106,2175,4293,8499,16864,33523,66727,

%T 132958,265137,529050,1056169,2109282,4213710,8419697,16827079,

%U 33634489,67237513,134424624,268768414,537407062,1074605619,2148875961,4297212424,8593556211,17185713097,34369170909

%N Sum of the smallest parts of all compositions of n.

%C Sums of the antidiagonals of A099238. - _Paul Barry_, Oct 08 2004

%H Alois P. Heinz, <a href="/A097939/b097939.txt">Table of n, a(n) for n = 1..1000</a> (correcting an earlier b-file from Vincenzo Librandi)

%F G.f.: Sum_{k>=1} x^k/(1-x-x^k).

%F a(n) = Sum_{r=0..n-1} Sum_{k=0..floor((n-r-1)/(r+1))} binomial(n-r(k+1)-1, k). - _Paul Barry_, Oct 08 2004

%F G.f.: (1-x)^2 * Sum_{k>=1} k*x^k/((x^k+x-1)*(x^(k+1)+x-1)). - _Vladeta Jovovic_, Apr 23 2006

%F G.f.: Sum_{k>=1} x^k/((1-x)^k*(1-x^k)). - _Vladeta Jovovic_, Mar 02 2008

%F G.f.: Sum_{n>=1} a*x^n/(1-a*x^n) (generalized Lambert series) where a=1/(1-x). - _Joerg Arndt_, Jan 30 2011

%F G.f.: Sum_{n>=1} (a*x)^n/(1-x^n) where a=1/(1-x). - _Joerg Arndt_, Jan 01 2013

%F G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1-x)^d. - _Paul D. Hanna_, Jul 18 2013

%F a(n) ~ 2^(n-1). - _Vaclav Kotesovec_, Oct 28 2014

%p A097939:=n->add(add(binomial(n-r*(k+1)-1,k), k=0..floor((n-r-1)/(r+1))), r=0..n-1): seq(A097939(n), n=1..50); # _Wesley Ivan Hurt_, Dec 03 2016

%p # second Maple Program:

%p b:= proc(n, m) option remember; `if`(n=0, m,

%p       add(b(n-j, min(j, m)), j=1..n))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=1..40);  # _Alois P. Heinz_, Jul 26 2020

%t Drop[ CoefficientList[ Series[ Sum[x^k/(1 - x - x^k), {k, 50}], {x, 0, 35}], x], 1] (* _Robert G. Wilson v_, Sep 08 2004 *)

%o (PARI)

%o N=66; x='x+O('x^N);

%o gf= sum(k=1,N, x^k/(1-x-x^k) );

%o Vec(gf)

%o /* _Joerg Arndt_, Jan 01 2013 */

%o (PARI) {a(n)=polcoeff(sum(m=1,n,x^m*sumdiv(m,d,1/(1-x +x*O(x^n))^d) ),n)}

%Y Cf. A046746, A092309, A097940, A097941, A099238, A336902, A336903.

%Y Column k=1 of A322427.

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Sep 05 2004

%E More terms from _Robert G. Wilson v_, Sep 08 2004