%I A119440
%S A119440 1,2,3,1,6,2,12,3,1,24,6,2,48,12,3,1,96,24,6,2,192,48,12,3,1,384,96,24,
%T A119440 6,2,768,192,48,12,3,1,1536,384,96,24,6,2,3072,768,192,48,12,3,1,6144,
%U A119440 1536,384,96,24,6,2,12288,3072,768,192,48,12,3,1,24576,6144,1536,384,96
%N A119440 Triangle read by rows: T(n,k) is the number of binary sequences of length 
               n that start with exactly k 01's (0<=k<=floor(n/2)).
%C A119440 Row n contains 1+floor(n/2) terms. Sum of entries in row n is 2^n (A000079). 
               T(n,0)=A098011(n+2). Except for a shift, all columns are identical. 
               G.f. of column k is x^(2k)*(1-x^2)/(1-2x). Sum(k*T(n,k),k=0..floor(n/
               2))=A000975(n-1).
%F A119440 T(n,k)=3*2^(n-2k-2) for n>=2k+2; T(2k,k)=1; T(2k+1,k)=2. G.f.=G(t,x)=(1-x^2)/
               [(1-2x)(1-tx^2)].
%e A119440 T(6,2)=3 because we have 010100, 010110 and 010111.
%e A119440 Triangle starts:
%e A119440 1;
%e A119440 2;
%e A119440 3,1;
%e A119440 6,2;
%e A119440 12,3,1;
%e A119440 24,6,2;
%e A119440 48,12,3,1;
%p A119440 T:=proc(n,k) if 2*k+2<=n then 3*2^(n-2*k-2) elif n=2*k then 1 elif n=2*k+1 
               then 2 else 0 fi end: for n from 0 to 16 do seq(T(n,k),k=0..floor(n/
               2)) od; # yields sequence in triangular form
%Y A119440 Cf. A000079, A098011, A000975.
%Y A119440 Sequence in context: A083855 A062565 A156344 this_sequence A165742 A162984 
               A166295
%Y A119440 Adjacent sequences: A119437 A119438 A119439 this_sequence A119441 A119442 
               A119443
%K A119440 nonn,tabf
%O A119440 0,2
%A A119440 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 19 2006