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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*q(n-1,x)+n+1, n>=0.
2

%I #7 Jan 03 2024 07:40:11

%S 1,1,2,1,3,5,2,5,9,13,3,8,14,21,28,5,13,23,34,46,58,8,21,37,55,74,94,

%T 114,13,34,60,89,120,152,185,218,21,55,97,144,194,246,299,353,407,34,

%U 89,157,233,314,398,484,571,659,747,55,144,254,377,508,644,783

%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*q(n-1,x)+n+1, n>=0.

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

%e First six rows:

%e 1

%e 1...2

%e 1...3....5

%e 2...5....9....13

%e 3...8....14...21...28

%e 5...13...23...34...46...58

%t z = 12;

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

%t q[n_, x_] := x*q[n - 1, x] + n + 1; q[0, x_] := 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}]] (* A193953 *)

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

%t Flatten[Table[g[n], {n, -1, z}]] (* A193954 *)

%Y Cf. A193722, A193954.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Aug 10 2011