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

%I A171682 #23 Jan 03 2024 06:49:05
%S A171682 1,2,3,6,10,20,37,72,140,275,540,1069,2118,4206,8365,16659,33204,
%T A171682 66231,132179,263913,527119,1053113,2104428,4205987,8407382,16807410,
%U A171682 33603024,67187111,134343790,268638648,537198557,1074270342,2148336463,4296343787,8592156886,17183457812,34365534564
%N A171682 Number of compositions of n with the smallest part in the first position.
%C A171682 First differences of A097939.
%H A171682 Alois P. Heinz, <a href="/A171682/b171682.txt">Table of n, a(n) for n = 1..1000</a>
%F A171682 G.f.: (1-x) * Sum_{k>=1} x^k/(1-x-x^k). [_Joerg Arndt_, Jan 01 2013]
%F A171682 a(n) ~ 2^(n-2). - _Vaclav Kotesovec_, Sep 10 2014
%F A171682 G.f.: Sum_{n>=1} q^n/(1-Sum_{k>=n} q^k). - _Joerg Arndt_, Jan 03 2024
%e A171682 The a(6)=20 such compositions of 6 are
%e A171682 [ 1]  [ 1 1 1 1 1 1 ]
%e A171682 [ 2]  [ 1 1 1 1 2 ]
%e A171682 [ 3]  [ 1 1 1 2 1 ]
%e A171682 [ 4]  [ 1 1 1 3 ]
%e A171682 [ 5]  [ 1 1 2 1 1 ]
%e A171682 [ 6]  [ 1 1 2 2 ]
%e A171682 [ 7]  [ 1 1 3 1 ]
%e A171682 [ 8]  [ 1 1 4 ]
%e A171682 [ 9]  [ 1 2 1 1 1 ]
%e A171682 [10]  [ 1 2 1 2 ]
%e A171682 [11]  [ 1 2 2 1 ]
%e A171682 [12]  [ 1 2 3 ]
%e A171682 [13]  [ 1 3 1 1 ]
%e A171682 [14]  [ 1 3 2 ]
%e A171682 [15]  [ 1 4 1 ]
%e A171682 [16]  [ 1 5 ]
%e A171682 [17]  [ 2 2 2 ]
%e A171682 [18]  [ 2 4 ]
%e A171682 [19]  [ 3 3 ]
%e A171682 [20]  [ 6 ]
%e A171682 - _Joerg Arndt_, Jan 01 2013.
%t A171682 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 *)
%o A171682 (PARI)
%o A171682 N=66; x='x+O('x^N);
%o A171682 gf= (1-x) * sum(k=1,N, x^k/(1-x-x^k) );
%o A171682 Vec(gf)
%o A171682 /* _Joerg Arndt_, Jan 01 2013 */
%Y A171682 Cf. A079500.
%K A171682 easy,nonn
%O A171682 1,2
%A A171682 _Vladeta Jovovic_, Dec 15 2009
%E A171682 Added more terms, _Joerg Arndt_, Jan 01 2013