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A193908 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+2)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2x*q(n-1,x)+1 with q(0,x)=1.

%I A193908 #5 Mar 30 2012 18:57:39
%S A193908 1,2,1,8,6,3,24,20,12,6,80,64,40,22,11,256,208,128,72,38,19,832,672,
%T A193908 416,232,124,64,32,2688,2176,1344,752,400,208,106,53,8704,7040,4352,
%U A193908 2432,1296,672,344,174,87,28160,22784,14080,7872,4192,2176,1112,564
%N A193908 Triangular array:  the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+2)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2x*q(n-1,x)+1 with q(0,x)=1.
%C A193908 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%C A193908 First five rows of P (triangle of coefficients of polynomials p(n,x)):
%C A193908 1
%C A193908 1...2
%C A193908 1...2...3
%C A193908 1...2...3...5
%C A193908 1...2...3...5...8
%C A193908 First five rows of Q:
%C A193908 1
%C A193908 2...1
%C A193908 4...2..1
%C A193908 8...4...2...1
%C A193908 16..8...4...2...1
%e A193908 First six rows:
%e A193908 1
%e A193908 2....1
%e A193908 8....6....3
%e A193908 24...20...12...6
%e A193908 80...64...40...22...11
%e A193908 256..208..128..72...38...19
%t A193908 z = 12;
%t A193908 p[n_, x_] := Sum[Fibonacci[k + 2]*x^(n - k), {k, 0, n}];
%t A193908 q[n_, x_] := 2 x*q[n - 1, x] + 1 ; q[0, x_] := 1;
%t A193908 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193908 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193908 g[n_] := CoefficientList[w[n, x], {x}]
%t A193908 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193908 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193908 *)
%t A193908 TableForm[Table[g[n], {n, -1, z}]]
%t A193908 Flatten[Table[g[n], {n, -1, z}]]  (* A193909 *)
%Y A193908 Cf. A193722, A193909.
%K A193908 nonn,tabl
%O A193908 0,2
%A A193908 _Clark Kimberling_, Aug 09 2011