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A118884 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).

%I A118884 #11 Dec 26 2013 08:29:15
%S A118884 1,2,4,8,15,1,28,4,52,12,96,32,177,78,1,326,180,6,600,400,24,1104,864,
%T A118884 80,2031,1827,237,1,3736,3800,648,8,6872,7800,1672,40,12640,15840,
%U A118884 4128,160,23249,31884,9846,556,1,42762,63704,22844,1752,10,78652,126480
%N A118884 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).
%C A118884 Row n has 1+floor(n/4) terms. Sum of entries in row n is 2^n (A000079). T(n,0) = A008937(n+1). T(n,1) = A118885(n). Sum(k*T(n,k), k=0..n-1) = (n-3)*2^(n-4) (A001787).
%H A118884 Alois P. Heinz, <a href="/A118884/b118884.txt">Rows n = 0..300, flattened</a>
%F A118884 G.f.: G(t,z) = 1/[1-2z+(1-t)z^4]. T(n,k) = 2T(n-1,k)-T(n-4,k)+T(n-4,k-1) (n>=4,k>=1).
%e A118884 T(9,2) = 6 because we have aa0, aa1, a0a, a1a, 0aa and 1aa, where a=0011.
%e A118884 Triangle starts:
%e A118884 1;
%e A118884 2;
%e A118884 4;
%e A118884 8;
%e A118884 15, 1;
%e A118884 28, 4;
%e A118884 52, 12;
%e A118884 96, 32;
%p A118884 G:=1/(1-2*z+(1-t)*z^4): Gser:=simplify(series(G,z=0,23)): P[0]:=1: for n from 1 to 19 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 19 do seq(coeff(P[n],t,j),j=0..floor(n/4)) od; # yields sequence in triangular form
%t A118884 nn=12;c=0;Map[Select[#,#>0&]&,CoefficientList[Series[1/(1-2x - (y-1)x^4/ (1-(y-1)c)), {x,0,nn}],{x,y}]]//Grid (* _Geoffrey Critzer_, Dec 25 2013 *)
%Y A118884 Cf. A000079, A008937, A118885, A001787.
%K A118884 nonn,tabf
%O A118884 0,2
%A A118884 _Emeric Deutsch_, May 03 2006