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A193790 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=1+x^n.

%I A193790 #4 Mar 30 2012 18:57:38
%S A193790 1,1,1,1,2,3,1,4,4,9,1,6,12,8,27,1,8,24,32,16,81,1,10,40,80,80,32,243,
%T A193790 1,12,60,160,240,192,64,729,1,14,84,280,560,672,448,128,2187,1,16,112,
%U A193790 448,1120,1792,1792,1024,256,6561,1,18,144,672,2016,4032,5376
%N A193790 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=1+x^n.
%C A193790 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%e A193790 First six rows:
%e A193790 1
%e A193790 1....1
%e A193790 1....2....3
%e A193790 1....4....4....9
%e A193790 1....6....12....8...27
%e A193790 1....8....24....32...16...81
%t A193790 z = 10; a = 2; b = 1;
%t A193790 p[n_, x_] := (a*x + b)^n
%t A193790 q[n_, x_] := 1 + x^n ; q[n_, 0] := q[n, x] /. x -> 0;
%t A193790 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193790 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193790 g[n_] := CoefficientList[w[n, x], {x}]
%t A193790 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193790 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193790 *)
%t A193790 TableForm[Table[g[n], {n, -1, z}]]
%t A193790 Flatten[Table[g[n], {n, -1, z}]]  (* A193791 *)
%Y A193790 Cf. A193791.
%K A193790 nonn,tabl
%O A193790 0,5
%A A193790 _Clark Kimberling_, Aug 05 2011