A168655 Number of compositions such that the number of parts is divisible by the first part.

1, 1, 3, 5, 11, 22, 44, 88, 177, 355, 710, 1419, 2838, 5679, 11363, 22727, 45443, 90862, 181703, 363419, 726903, 1453875, 2907667, 5814880, 11628864, 23256828, 46513965, 93031069, 186068503, 372142797, 744280096, 1488527555, 2976987042, 5953897971, 11907811651

Offset: 1

Author

Vladeta Jovovic, Dec 01 2009

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A079501.

Programs

Maple

b:= proc(n,t,g) option remember; `if`(n=0,
      `if`(irem(t, g)=0, 1, 0), add(b(n-i, t+1,
      `if`(g=0,i,g)), i=1..n))
    end:
a:= n-> b(n,0,0):
seq(a(n), n=1..40); # Alois P. Heinz, Dec 15 2009

Mathematica

A101510[n_] := Sum[If[Mod[i+1, k+1] == 0, Binomial[n-k, i], 0], {k, 0, n/2}, {i, 0, n-k}]; A168655 =  Join[{1}, Table[A101510[n], {n, 0, 32}] // Differences] (* Jean-François Alcover, Jan 24 2014 *)

Formula

G.f.: (1-x)*Sum(x^(2*n-1)/((1-x)^n-x^n),n=1..infinity), First differences of A101510.

a(n) ~ log(2) * 2^(n-1). - Vaclav Kotesovec, May 01 2014

Extensions

More terms from Alois P. Heinz, Dec 15 2009