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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.)
2

%I #7 Feb 19 2015 14:24:45

%S 0,1,1,9,17,80,198,748,2107,7236,21680,71279,219879,708436,2215513,

%T 7071210,22256567,70723367,223272153,708017329,2238347440,7091170416,

%U 22433032016

%N 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 The definition of Q-residue is given at A193649.

%F 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 q[n_, k_] := k + 1;

%t r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];

%t f[n_, x_] := Fibonacci[n, x]; (* A049310 *)

%t p[n_, k_] := Coefficient[f[n, x], x, k];

%t v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]

%t Table[v[n], {n, 0, 22}] (* A193663 *)

%t TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]

%t Table[r[k], {k, 0, 8}]

%t TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]]

%Y Cf. A049310, A193649.

%K nonn

%O 0,4

%A _Clark Kimberling_, Aug 02 2011