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A193821 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

%I A193821 #5 Mar 30 2012 18:57:38
%S A193821 1,1,1,2,3,3,4,8,9,9,8,20,26,27,27,16,48,72,80,81,81,32,112,192,232,
%T A193821 242,243,243,64,256,496,656,716,728,729,729,128,576,1248,1808,2088,
%U A193821 2172,2186,2187,2187,256,1280,3072,4864,5984,6432,6544,6560,6561,6561
%N A193821 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=x^n+x^(n-1)+...+x+1.
%C A193821 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%e A193821 First six rows:
%e A193821 1
%e A193821 1....1
%e A193821 2....3....3
%e A193821 4....8....9....9
%e A193821 8....20...26...27...27
%e A193821 16...48...72...80...81...81
%t A193821 p[n_, x_] := (a*x + b)^n
%t A193821 q[0, x_] := 1
%t A193821 q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
%t A193821 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193821 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193821 g[n_] := CoefficientList[w[n, x], {x}]
%t A193821 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193821 Flatten[Table[Reverse[g[n]], {n, -1, z}]]   (* A193821 *)
%t A193821 TableForm[Table[g[n], {n, -1, z}]]
%t A193821 Flatten[Table[g[n], {n, -1, z}]]  (* A193822 *)
%Y A193821 Cf. A193722, A193822.
%K A193821 nonn,tabl
%O A193821 0,4
%A A193821 _Clark Kimberling_, Aug 06 2011