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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+1)^n.
2

%I #5 Mar 30 2012 18:57:39

%S 1,1,1,1,3,2,2,7,9,4,3,14,25,21,7,5,28,64,75,46,12,8,53,148,224,195,

%T 94,20,13,99,326,603,679,468,185,33,21,181,687,1502,2073,1855,1056,

%U 353,54,34,327,1405,3543,5786,6357,4711,2280,659,88,55,584,2802,8005

%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+1)^n.

%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...3....2

%e 2...7....9....4

%e 3...14...25...21...7

%e 5...28...64...75...46...12

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

%t q[n_, x_] := (x + 1)^n;

%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}]] (* A193919 *)

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

%t Flatten[Table[g[n], {n, -1, z}]] (* A193920 *)

%Y Cf. A193722, A193920.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Aug 09 2011