A097979 - OEIS

%I #40 Oct 16 2024 14:24:58

%S 1,3,6,12,23,46,91,183,367,737,1478,2962,5928,11858,23707,47384,94698,

%T 189260,378277,756160,1511730,3022672,6044472,12088395,24177600,

%U 48359695,96732370,193495606,387057584,774248858,1548754115,3097980230,6196797193,12395022288

%N Total number of largest parts in all compositions of n.

%C Also number of compositions of n+1 with unique largest part. - _Vladeta Jovovic_, Apr 03 2005

%H Alois P. Heinz, <a href="/A097979/b097979.txt">Table of n, a(n) for n = 1..1000</a> (first 200 terms from Vincenzo Librandi)

%F G.f.: (1-x)^2 * Sum_{k>=1} x^k/(1-2*x+x^(k+1))^2.

%F a(n) ~ 2^(n-1)/log(2). - _Vaclav Kotesovec_, Apr 30 2014

%t nn=32; Drop[CoefficientList[Series[Sum[x^j/(1 - (x - x^(j + 1))/(1 - x))^2, {j, 1, nn}], {x, 0, nn}], x], 1] (* _Geoffrey Critzer_, Mar 31 2014 *)

%t b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_, k_] := Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]; a[0, 0] = 1; a[_, 0] = 0; a[n_] := a[n+1, 1]; Table[a[n], {n, 1, 32}] (* _Jean-François Alcover_, Feb 10 2015, after A238341 *)

%o (PARI) { b(t)=local(r);sum(k=1,t, forstep(s=t%k,t-k,k,u=(t-k-s)\k;r+=binomial(-2,s)*(-2)^(s-u)*binomial(s,u)));r }

%o { a(n)=b(n)-2*b(n-1)+b(n-2) } \\ _Max Alekseyev_, Apr 16 2005

%Y Cf. A097941, A046746.

%Y Column k=1 of A238341.

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Sep 07 2004

%E More terms from _Max Alekseyev_, Apr 16 2005