A097939 Sum of the smallest parts of all compositions of n.
1, 3, 6, 12, 22, 42, 79, 151, 291, 566, 1106, 2175, 4293, 8499, 16864, 33523, 66727, 132958, 265137, 529050, 1056169, 2109282, 4213710, 8419697, 16827079, 33634489, 67237513, 134424624, 268768414, 537407062, 1074605619, 2148875961, 4297212424, 8593556211, 17185713097, 34369170909
Offset: 1
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000 (correcting an earlier b-file from Vincenzo Librandi)
Programs
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Maple
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 # second Maple Program: b:= proc(n, m) option remember; `if`(n=0, m, add(b(n-j, min(j, m)), j=1..n)) end: a:= n-> b(n$2): seq(a(n), n=1..40); # Alois P. Heinz, Jul 26 2020
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Mathematica
Drop[ CoefficientList[ Series[ Sum[x^k/(1 - x - x^k), {k, 50}], {x, 0, 35}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *) -
PARI
N=66; x='x+O('x^N); gf= sum(k=1,N, x^k/(1-x-x^k) ); Vec(gf) /* Joerg Arndt, Jan 01 2013 */ -
PARI
{a(n)=polcoeff(sum(m=1,n,x^m*sumdiv(m,d,1/(1-x +x*O(x^n))^d) ),n)}
Formula
G.f.: Sum_{k>=1} x^k/(1-x-x^k).
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
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
G.f.: Sum_{k>=1} x^k/((1-x)^k*(1-x^k)). - Vladeta Jovovic, Mar 02 2008
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
G.f.: Sum_{n>=1} (a*x)^n/(1-x^n) where a=1/(1-x). - Joerg Arndt, Jan 01 2013
G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1-x)^d. - Paul D. Hanna, Jul 18 2013
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Oct 28 2014
Extensions
More terms from Robert G. Wilson v, Sep 08 2004
Comments