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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)=x^n+x^(n-1)+...+x+1.
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%I #5 Mar 30 2012 18:57:39

%S 1,1,1,1,2,2,2,3,4,4,3,5,6,7,7,5,8,10,11,12,12,8,13,16,18,19,20,20,13,

%T 21,26,29,31,32,33,33,21,34,42,47,50,52,53,54,54,34,55,68,76,81,84,86,

%U 87,88,88,55,89,110,123,131,136,139,141,142,143,143,89,144,178

%N 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)=x^n+x^(n-1)+...+x+1.

%C See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

%e First six rows:

%e 1

%e 1...1

%e 1...2...2

%e 2...3...4....4

%e 3...5...6....7....7

%e 5...8...10...11...12...12

%t p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];

%t q[n_, x_] := x*q[n - 1, x] + 1; q[0, n_] := 1;

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193921 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A193922 *)

%Y Cf. A193722, A193922.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Aug 09 2011