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A102712 Sum of largest parts of all compositions of n.

%I A102712 #34 Aug 10 2020 02:28:14
%S A102712 1,3,8,19,43,94,202,428,899,1875,3890,8036,16544,33962,69552,142149,
%T A102712 290017,590814,1202016,2442706,4958974,10058216,20384498,41282346,
%U A102712 83549603,168992081,341627732,690279026,1394115072,2814430326,5679552630,11457287926,23104929222
%N A102712 Sum of largest parts of all compositions of n.
%H A102712 Alois P. Heinz and Vaclav Kotesovec, <a href="/A102712/b102712.txt">Table of n, a(n) for n = 1..3000</a> (first 1000 terms from Alois P. Heinz)
%F A102712 G.f.: Sum(n*(1-x)^2*x^n/((1-2*x+x^n)*(1-2*x+x^(n+1))), n=1..infinity).
%F A102712 G.f.: (1-x)/(1-2*x)*Sum(x^n/(1-2*x+x^n),n=1..infinity). - _Vladeta Jovovic_, Apr 28 2008
%e A102712 a(4) = 19 because we have (4), (3)1, 1(3), (2)2, (2)11, 1(2)1, 11(2) and (1)111; the largest parts, shown between parentheses, add up to 19.
%p A102712 G:=sum(n*(1-x)^2*x^n/((1-2*x+x^n)*(1-2*x+x^(n+1))),n=1..45): Gser:=series(G,x=0,40): seq(coeff(Gser,x^n),n=1..36); # _Emeric Deutsch_, Mar 29 2005
%p A102712 # second Maple program:
%p A102712 b:= proc(n, m, t) option remember;
%p A102712       `if`(m=1, 1,
%p A102712       `if`(n<m and not t, 0,
%p A102712       `if`(n=0, 1, add(b(n-j, m, j=m or t), j=1..min(n, m)))))
%p A102712     end:
%p A102712 a:= n-> add(m*b(n, m, false), m=1..n):
%p A102712 seq(a(n), n=1..40);  # _Alois P. Heinz_, Oct 21 2011
%t A102712 nn=33;f[list_]:=Sum[list[[i]]i,{i,1,Length[list]}];Drop[Map[f,Transpose[Table[CoefficientList[Series[1/(1-(x-x^(k+1))/(1-x))-1/(1-(x-x^k)/(1-x)),{x,0,nn}],x],{k,1,nn}]]],1] (* _Geoffrey Critzer_, Apr 06 2014 *)
%Y A102712 Cf. A006128, A097939, A336902, A336903.
%Y A102712 Column k=1 of A322428.
%K A102712 easy,nonn
%O A102712 1,2
%A A102712 _Vladeta Jovovic_, Feb 05 2005
%E A102712 More terms from _Emeric Deutsch_, Mar 29 2005