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Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(3x+1)^n and q(n,x)=1+x^n.
2

%I #5 Mar 30 2012 18:57:38

%S 1,1,1,1,3,4,1,6,9,16,1,9,27,27,64,1,12,54,108,81,256,1,15,90,270,405,

%T 243,1024,1,18,135,540,1215,1458,729,4096,1,21,189,945,2835,5103,5103,

%U 2187,16384,1,24,252,1512,5670,13608,20412,17496,6561,65536,1

%N Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(3x+1)^n and q(n,x)=1+x^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....4

%e 1...6....9....16

%e 1...9....27...27....64

%e 1...12...54...108...81...256

%t z = 9; a = 3; b = 1;

%t p[n_, x_] := (a*x + b)^n

%t q[n_, x_] := 1 + x^n ; q[n_, 0] := q[n, x] /. x -> 0;

%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}]] (* A193794 *)

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

%t Flatten[Table[g[n], {n, -1, z}]] (* A193795 *)

%Y Cf. A193722, A193795.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Aug 05 2011