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A193663 Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q is the triangle given by t(n,k)=k+1 for 0<=k<=n. (See Comments.)

%I A193663 #7 Feb 19 2015 14:24:45
%S A193663 0,1,1,9,17,80,198,748,2107,7236,21680,71279,219879,708436,2215513,
%T A193663 7071210,22256567,70723367,223272153,708017329,2238347440,7091170416,
%U A193663 22433032016
%N A193663 Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q is the triangle given by t(n,k)=k+1 for 0<=k<=n.  (See Comments.)
%C A193663 The definition of Q-residue is given at A193649.
%F A193663 Conjecture: G.f.: x*(1-x+x^2) / ( 1-2*x-6*x^2+7*x^3+x^4 ). - _R. J. Mathar_, Feb 19 2015
%t A193663 q[n_, k_] := k + 1;
%t A193663 r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
%t A193663 f[n_, x_] := Fibonacci[n, x]; (* A049310 *)
%t A193663 p[n_, k_] := Coefficient[f[n, x], x, k];
%t A193663 v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
%t A193663 Table[v[n], {n, 0, 22}]    (* A193663 *)
%t A193663 TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
%t A193663 Table[r[k], {k, 0, 8}]
%t A193663 TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]
%Y A193663 Cf. A049310, A193649.
%K A193663 nonn
%O A193663 0,4
%A A193663 _Clark Kimberling_, Aug 02 2011