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A105114 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 2.

%I A105114 #10 Nov 05 2012 19:33:15
%S A105114 1,1,1,1,2,2,4,3,1,7,6,3,12,13,6,1,21,26,13,4,37,50,30,10,1,65,96,66,
%T A105114 24,5,114,184,139,59,15,1,200,350,288,140,40,6,351,661,591,318,105,21,
%U A105114 1,616,1242,1199,704,266,62,7,1081,2324,2406,1533,645,174,28,1,1897,4332
%N A105114 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 2.
%C A105114 Row n has 1+floor(n/2) terms. Row sums are the powers of 2 (A000079). Column 0 yields A005251.
%C A105114 Number of binary words of length n-1 having k isolated 0's. Example: T(5,1)=6 because we have 0111, 0100, 1011, 1101, 0010 and 1110. - _Emeric Deutsch_, May 21 2006
%H A105114 Alois P. Heinz, <a href="/A105114/b105114.txt">Rows n = 0..200, flattened</a>
%F A105114 G.f.: (1-z)/(1-2z+z^2-z^3-tz^2+tz^3).
%e A105114 T(7,3) = 4 because we have (1,2,2,2), (2,1,2,2), (2,2,1,2) and (2,2,2,1).
%e A105114 Triangle begins:
%e A105114 1;
%e A105114 1;
%e A105114 1,      1;
%e A105114 2,      2;
%e A105114 4,      3,    1;
%e A105114 7,      6,    3;
%e A105114 12,    13,    6,   1;
%e A105114 21,    26,   13,   4;
%e A105114 37,    50,   30,  10,   1;
%e A105114 65,    96,   66,  24,   5;
%e A105114 114,  184,  139,  59,  15,  1;
%e A105114 200,  350,  288, 140,  40,  6;
%e A105114 351,  661,  591, 318, 105, 21,  1;
%e A105114 616, 1242, 1199, 704, 266, 62,  7;
%p A105114 G:=(1-z)/(1-2*z-z^2*t+z^3*t+z^2-z^3):Gser:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 16 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od;# yields sequence in triangular form
%t A105114 nn=15;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[1/(1-(x/(1-x)-x^2+y x^2)),{x,0,nn}],{x,y}]]//Grid  (* _Geoffrey Critzer_, Nov 05 2012 *)
%Y A105114 Cf. A000079, A005251.
%K A105114 nonn,tabf
%O A105114 0,5
%A A105114 _Emeric Deutsch_, Apr 07 2005