A102712 Sum of largest parts of all compositions of n.

1, 3, 8, 19, 43, 94, 202, 428, 899, 1875, 3890, 8036, 16544, 33962, 69552, 142149, 290017, 590814, 1202016, 2442706, 4958974, 10058216, 20384498, 41282346, 83549603, 168992081, 341627732, 690279026, 1394115072, 2814430326, 5679552630, 11457287926, 23104929222

Offset: 1

Author

Vladeta Jovovic, Feb 05 2005

Keywords

• easy
• nonn

Examples

			a(4) = 19 because we have (4), (3)1, 1(3), (2)2, (2)11, 1(2)1, 11(2) and (1)111; the largest parts, shown between parentheses, add up to 19.
		

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..3000 (first 1000 terms from Alois P. Heinz)

Crossrefs

Cf. A006128, A097939, A336902, A336903.

Column k=1 of A322428.

Programs

Maple

G:=sum(n*(1-x)^2*x^n/((1-2*x+x^n)*(1-2*x+x^(n+1))),n=1..45): Gser:=series(G,x=0,40): seq(coeff(Gser,x^n),n=1..36); # Emeric Deutsch, Mar 29 2005
# second Maple program:
b:= proc(n, m, t) option remember;
      `if`(m=1, 1,
      `if`(n add(m*b(n, m, false), m=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 21 2011

Mathematica

nn=33;f[list_]:=Sum[list[[i]]i,{i,1,Length[list]}];Drop[Map[f,Transpose[Table[CoefficientList[Series[1/(1-(x-x^(k+1))/(1-x))-1/(1-(x-x^k)/(1-x)),{x,0,nn}],x],{k,1,nn}]]],1] (* Geoffrey Critzer, Apr 06 2014 *)

Formula

G.f.: Sum(n*(1-x)^2*x^n/((1-2*x+x^n)*(1-2*x+x^(n+1))), n=1..infinity).

G.f.: (1-x)/(1-2*x)*Sum(x^n/(1-2*x+x^n),n=1..infinity). - Vladeta Jovovic, Apr 28 2008

Extensions

More terms from Emeric Deutsch, Mar 29 2005