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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)(n+1)*x^(n-k) : 0<=k<=n}.
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%I #6 Mar 30 2012 18:57:39

%S 1,2,4,3,8,13,8,19,32,45,15,38,64,92,120,30,75,128,184,242,300,56,142,

%T 243,352,464,578,692,104,264,454,659,872,1088,1306,1524,189,482,831,

%U 1210,1604,2006,2411,2818,3225,340,869,1502,2191,2910,3644,4386

%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)(n+1)*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 2....4

%e 3....8....13

%e 8....19...32...45

%e 15...38...64...92...120

%e 30...75...128..184..242..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) (n + 1)*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}]] (* A193949 *)

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

%t Flatten[Table[g[n], {n, -1, z}]] (* A193950 *)

%Y Cf. A193722, A193950.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 10 2011