A097979 Total number of largest parts in all compositions of n.

1, 3, 6, 12, 23, 46, 91, 183, 367, 737, 1478, 2962, 5928, 11858, 23707, 47384, 94698, 189260, 378277, 756160, 1511730, 3022672, 6044472, 12088395, 24177600, 48359695, 96732370, 193495606, 387057584, 774248858, 1548754115, 3097980230, 6196797193, 12395022288

Offset: 1

Author

Vladeta Jovovic, Sep 07 2004

Keywords

• easy
• nonn

Comments

Also number of compositions of n+1 with unique largest part. - Vladeta Jovovic, Apr 03 2005

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from Vincenzo Librandi)

Crossrefs

Cf. A097941, A046746.

Column k=1 of A238341.

Programs

Mathematica

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 *)
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 *)

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 }
{ a(n)=b(n)-2*b(n-1)+b(n-2) } \\ Max Alekseyev, Apr 16 2005

Formula

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

a(n) ~ 2^(n-1)/log(2). - Vaclav Kotesovec, Apr 30 2014

Extensions

More terms from Max Alekseyev, Apr 16 2005