%N Triangle read by rows: T(n,k) is the coefficient of t^k (k>=1) in the polynomial P[n,t] defined by P[1,t]=P[2,t]=t^2, P[n,t]=tP[n-1,t]+t^2*P^2[n-2,t] (n>=3).
%C T(n,k) is the number of certain types of trees (see the Duke et al. reference) of height n and having k edges. Row n contains A027383(n-1) terms, the first n-1 of which are 0. Row sums yield A000278.
%H W. Duke, Stephen J. Greenfield and Eugene R. Speer, <a href="https://cs.uwaterloo.ca/journals/JIS/green2/qf.html">Properties of a Quadratic Fibonacci Recurrence</a>, J. Integer Sequences, 1998, #98.1.8.
%e P[3,t]=t^3+t^4; therefore T(3,1)=0, T(3,2)=0, T(3,3)=1, T(3,4)=1.
%e Triangle begins:
%p P:=t:P:=t^2:for n from 3 to 12 do P[n]:=sort(expand(t*P[n-1]+t^2*P[n-2]^2)) od: d:=1: d:=2: for n from 3 to 20 do d[n]:=2*d[n-2]+2 od: for n from 1 to 9 do seq(coeff(P[n],t^k),k=1..d[n]) od;# yields sequence in triangular form
%Y Cf. A000278, A027383.
%A _Emeric Deutsch_, Mar 21 2005