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

1,2

Links

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

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

Example

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.

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<m and not t, 0,
      `if`(n=0, 1, add(b(n-j, m, j=m or t), j=1..min(n, m)))))
    end:
a:= 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 *)

Crossrefs

Cf. A006128, A097939, A336902, A336903.

Column k=1 of A322428.

Sequence in context: A360489 A259401 A008466 * A054480 A371796 A191787

Adjacent sequences: A102709 A102710 A102711 * A102713 A102714 A102715

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 05 2005

Extensions

More terms from Emeric Deutsch, Mar 29 2005

Status

approved