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A178522 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).

%I A178522 #20 Jan 05 2025 19:51:39
%S A178522 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 A178522 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 A178522 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 A178522 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 A178522 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 A178522 Sum of entries in row n is A001595(n).
%C A178522 Sum_{k=0..n-1} k*T(n,k) = A178523(n).
%D A178522 D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
%H A178522 Y. Horibe, <a href="https://web.archive.org/web/2024*/https://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 A178522 G.f.: G(t,z)=(1-tz+tz^2)/[(1-z)(1-tz-tz^2)].
%F A178522 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 A178522 Triangle starts:
%e A178522 1,
%e A178522 1,
%e A178522 1,2,
%e A178522 1,2,2,
%e A178522 1,2,4,2,
%e A178522 1,2,4,6,2,
%e A178522 1,2,4,8,8,2,
%e A178522 1,2,4,8,14,10,2,
%e A178522 1,2,4,8,16,22,12,2,
%e A178522 1,2,4,8,16,30,32,14,2,
%e A178522 ...
%p A178522 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 A178522 Cf. A001595, A059214, A178523, A067331, A002940. See A059250 for another version.
%K A178522 nonn,tabf
%O A178522 0,4
%A A178522 _Emeric Deutsch_, Jun 15 2010