A171682 Number of compositions of n with the smallest part in the first position.

1, 2, 3, 6, 10, 20, 37, 72, 140, 275, 540, 1069, 2118, 4206, 8365, 16659, 33204, 66231, 132179, 263913, 527119, 1053113, 2104428, 4205987, 8407382, 16807410, 33603024, 67187111, 134343790, 268638648, 537198557, 1074270342, 2148336463, 4296343787, 8592156886, 17183457812, 34365534564

Offset: 1

Author

Vladeta Jovovic, Dec 15 2009

Keywords

• easy
• nonn

Comments

First differences of A097939.

Examples

			The a(6)=20 such compositions of 6 are
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 1 3 ]
[ 5]  [ 1 1 2 1 1 ]
[ 6]  [ 1 1 2 2 ]
[ 7]  [ 1 1 3 1 ]
[ 8]  [ 1 1 4 ]
[ 9]  [ 1 2 1 1 1 ]
[10]  [ 1 2 1 2 ]
[11]  [ 1 2 2 1 ]
[12]  [ 1 2 3 ]
[13]  [ 1 3 1 1 ]
[14]  [ 1 3 2 ]
[15]  [ 1 4 1 ]
[16]  [ 1 5 ]
[17]  [ 2 2 2 ]
[18]  [ 2 4 ]
[19]  [ 3 3 ]
[20]  [ 6 ]
- _Joerg Arndt_, Jan 01 2013.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A079500.

Programs

Mathematica

nn=37;Drop[CoefficientList[Series[Sum[x^i/(1-x^i/(1-x)),{i,1,nn}],{x,0,nn}],x],1]  (* Geoffrey Critzer, Mar 12 2013 *)

PARI

N=66; x='x+O('x^N);
gf= (1-x) * sum(k=1,N, x^k/(1-x-x^k) );
Vec(gf)
/* Joerg Arndt, Jan 01 2013 */

Formula

G.f.: (1-x) * Sum_{k>=1} x^k/(1-x-x^k). [Joerg Arndt, Jan 01 2013]

a(n) ~ 2^(n-2). - Vaclav Kotesovec, Sep 10 2014

G.f.: Sum_{n>=1} q^n/(1-Sum_{k>=n} q^k). - Joerg Arndt, Jan 03 2024

Extensions

Added more terms, Joerg Arndt, Jan 01 2013