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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)=sum{((k+1)^2)*x^(n-k) : 0<=k<=n}.
3

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

%S 1,1,4,1,5,13,2,9,23,45,3,14,36,71,120,5,23,59,116,196,300,8,37,95,

%T 187,316,484,692,13,60,154,303,512,784,1121,1524,21,97,249,490,828,

%U 1268,1813,2465,3225,34,157,403,793,1340,2052,2934,3989,5219,6625,55

%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)=sum{((k+1)^2)*x^(n-k) : 0<=k<=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...4

%e 1...5....13

%e 2...9....23...45

%e 3...14...36...71....120

%e 5...23...59...116...196...300

%t z = 12;

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

%t q[n_, x_] := Sum[((k + 1)^2)*x^(n - k), {k, 0, 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}]] (* A193955 *)

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

%t Flatten[Table[g[n], {n, -1, z}]] (* A193956 *)

%Y Cf. A193722, A193956.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Aug 10 2011