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%I #18 Jan 29 2016 05:20:07
%S 1,1,1,2,1,2,2,1,2,4,2,1,2,4,6,2,1,2,4,8,8,2,1,2,4,8,14,10,2,1,2,4,8,
%T 16,22,12,2,1,2,4,8,16,30,32,14,2,1,2,4,8,16,32,52,44,16,2,1,2,4,8,16,
%U 32,62,84,58,18,2,1,2,4,8,16,32,64,114,128,74,20,2,1,2,4,8,16,32,64,126
%N Triangle read by rows: T(n,k) is the number of nodes at level k in the Fibonacci tree of order n (n>=0, 0<=k<=n-1).
%C A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.
%C Sum of entries in row n is A001595(n).
%C Sum_{k=0..n-1} k*T(n,k) = A178523(n).
%D D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
%H Y. Horibe, <a href="http://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
%F G.f.: G(t,z)=(1-tz+tz^2)/[(1-z)(1-tz-tz^2)].
%F T(k,n) = T(k-1,n-1)+T(k-1,n) with T(0,0)=1, T(k,0)=1 for k>0, T(0,n)=2 for n>0. - _Frank M Jackson_, Aug 30 2011
%e Triangle starts:
%e 1,
%e 1,
%e 1,2,
%e 1,2,2,
%e 1,2,4,2,
%e 1,2,4,6,2,
%e 1,2,4,8,8,2,
%e 1,2,4,8,14,10,2,
%e 1,2,4,8,16,22,12,2,
%e 1,2,4,8,16,30,32,14,2,
%e ...
%p G := (1-t*z+t*z^2)/((1-z)*(1-t*z-t*z^2)): Gser := simplify(series(G, z = 0, 17)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 13 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form
%Y Cf. A001595, A059214, A178523, A067331, A002940. See A059250 for another version.
%K nonn,tabf
%O 0,4
%A _Emeric Deutsch_, Jun 15 2010
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