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A193915 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*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 A193915 #6 Mar 30 2012 18:57:39
%S A193915 1,2,1,4,4,2,16,12,8,4,48,40,24,14,7,160,128,80,44,24,12,512,416,256,
%T A193915 144,76,40,20,1664,1344,832,464,248,128,66,33,5376,4352,2688,1504,800,
%U A193915 416,212,108,54,17408,14080,8704,4864,2592,1344,688,348,176,88
%N A193915 Triangular array:  the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*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 A193915 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%C A193915 First five rows of P (triangle of coefficients of polynomials p(n,x)):
%C A193915 1
%C A193915 1...1
%C A193915 1...1...2
%C A193915 1...1...2...3
%C A193915 1...1...2...3...5
%C A193915 First five rows of Q:
%C A193915 1
%C A193915 2....1
%C A193915 4....2...1
%C A193915 8....4...2...1
%C A193915 16...8...4...2...1
%e A193915 First six rows:
%e A193915 1
%e A193915 2....1
%e A193915 4....4....2
%e A193915 16...12...8...4
%e A193915 48...40...24..14..7
%e A193915 160..128..80..44..24..12
%t A193915 z = 12;
%t A193915 p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t A193915 q[n_, x_] := 2 x*q[n - 1, x] + 1 ; q[0, x_] := 1;
%t A193915 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193915 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193915 g[n_] := CoefficientList[w[n, x], {x}]
%t A193915 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193915 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193915 *)
%t A193915 TableForm[Table[g[n], {n, -1, z}]]
%t A193915 Flatten[Table[g[n], {n, -1, z}]]  (* A193916 *)
%Y A193915 Cf. A193722, A193916, A193908.
%K A193915 nonn,tabl
%O A193915 0,2
%A A193915 _Clark Kimberling_, Aug 09 2011