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Search: id:A129706
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| A129706 |
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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. |
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+0
2
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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).
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FORMULA
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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.
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EXAMPLE
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T(5,3)=4 because we have 11101, 10101, 01110 and 01010.
Triangle starts:
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;
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MAPLE
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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
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CROSSREFS
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Cf. A000045, A129707.
Sequence in context: A119646 A024693 A025126 this_sequence A160384 A024327 A073044
Adjacent sequences: A129703 A129704 A129705 this_sequence A129707 A129708 A129709
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 12 2007
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