%I A105039
%S A105039 1,1,3,3,8,11,20,34,59,96,167,282,475,800,1352,2275,3828,6426,10785,
%T A105039 18085,30297,50698,84770,141623,236425,394381,657380,1094975,1822628,
%U A105039 3031843,5040129,8373594,13903588,23072567,38267330,63435438,105103059
%N A105039 Number of compositions of n with unique smallest part.
%F A105039 G.f.: Sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)), k=1..infinity).
%F A105039 Also (1-x)^2*Sum(x^k/(1-x-x^(k+1))^2, k=1..infinity). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Apr 05 2005
%F A105039 a(n) = 1 + sum(k=2..[(n+3)/2], k * sum(s=1..[(n-1)/k], binomial(n-k*s-1, 
               k-2) ) ) - Max Alekseyev (maxale(AT)gmail.com), Apr 15 2005
%e A105039 a(5)=8 because we have 5,14,41,23,32,122,212 and 221.
%p A105039 G:=sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)),k=1..70):Gser:=series(G,x=0,
               44):seq(coeff(Gser,x^n),n=1..41); (Deutsch)
%o A105039 (PARI) a(n)=1+sum(k=2,(n+3)\2,k*sum(s=1,(n-1)\k,binomial(n-k*s-1,k-2))) 
               (Alekseyev)
%Y A105039 Cf. A079501, A097979.
%Y A105039 Sequence in context: A141577 A123315 A052407 this_sequence A090597 A126073 
               A126592
%Y A105039 Adjacent sequences: A105036 A105037 A105038 this_sequence A105040 A105041 
               A105042
%K A105039 easy,nonn
%O A105039 1,3
%A A105039 Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 03 2005
%E A105039 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu) and Max Alekseyev 
               (maxale(AT)gmail.com), Apr 13 2005