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