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A129706 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k inversions (n>=0, 0<=k<=floor(n(n+1)/6)). A Fibonacci binary word is a binary word having no 00 subword.

%I A129706 #3 Mar 30 2012 17:36:14
%S A129706 1,2,2,1,2,2,1,2,2,2,2,2,2,2,4,2,1,2,2,2,4,4,4,2,1,2,2,2,4,4,6,6,4,2,
%T A129706 2,2,2,2,4,4,6,8,8,6,6,4,2,1,2,2,2,4,4,6,8,10,10,10,10,8,6,4,2,1,2,2,
%U A129706 2,4,4,6,8,10,12,14,14,14,14,12,10,8,4,2,2,2,2,2,4,4,6,8,10,12,16,18,18,20
%N A129706 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k inversions (n>=0, 0<=k<=floor(n(n+1)/6)). A Fibonacci binary word is a binary word having no 00 subword.
%C A129706 Row n has 1+floor(n(n+1)/6) terms. Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), k>=0)=A129707(n).
%F A129706 G.f.=G(t,z)=H(t,1,z), where H(t,x,z)=1+z+xzH(t,x,z)+txz^2*H(t,tx,z). Row generating polynomials P[n] are given by P[n](t)=Q[n](t,1), where Q[0]=1, Q[1]=1+x, Q[n](t,x)=xQ[n-1](t,x)+txQ[n-2](t,tx) for n>=2.
%e A129706 T(5,3)=4 because we have 11101, 10101, 01110 and 01010.
%e A129706 Triangle starts:
%e A129706 1;
%e A129706 2;
%e A129706 2,1;
%e A129706 2,2,1;
%e A129706 2,2,2,2;
%e A129706 2,2,2,4,2,1;
%e A129706 2,2,2,4,4,4,2,1;
%p A129706 Q[0]:=1: Q[1]:=1+x: for n from 2 to 12 do Q[n]:=expand(x*Q[n-1]+t*x*subs(x=t*x,Q[n-2])) od: for n from 0 to 15 do P[n]:=sort(subs(x=1,Q[n])) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..floor(n*(n+1)/6)) od; # yields sequence in triangular form
%Y A129706 Cf. A000045, A129707.
%K A129706 nonn,tabf
%O A129706 0,2
%A A129706 _Emeric Deutsch_, May 12 2007