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A193724 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+1)^n.

%I A193724 #20 Jan 22 2020 15:21:56
%S A193724 1,1,1,2,5,3,4,16,21,9,8,44,90,81,27,16,112,312,432,297,81,32,272,960,
%T A193724 1800,1890,1053,243,64,640,2736,6480,9180,7776,3645,729,128,1472,7392,
%U A193724 21168,37800,43092,30618,12393,2187,256,3328,19200,64512,139104
%N A193724 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+1)^n.
%C A193724 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%C A193724 Triangle T(n,k), read by rows, given by [1,1,0,0,0,0,0,0,0,...] DELTA [1,2,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 04 2011
%F A193724 T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,0)=T(1,1)=1. - _Philippe Deléham_, Oct 05 2011
%F A193724 G.f.: (-1+x+2*x*y)/(-1+2*x+3*x*y). - _R. J. Mathar_, Aug 11 2015
%e A193724 First six rows:
%e A193724    1;
%e A193724    1,   1;
%e A193724    2,   5,   3;
%e A193724    4,  16,  21,   9;
%e A193724    8,  44,  90,  81,  27;
%e A193724   16, 112, 312, 432, 297, 81;
%t A193724 z = 8; a = 1; b = 2; c = 1; d = 1;
%t A193724 p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t A193724 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193724 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193724 g[n_] := CoefficientList[w[n, x], {x}]
%t A193724 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193724 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193724 *)
%t A193724 TableForm[Table[g[n], {n, -1, z}]]
%t A193724 Flatten[Table[g[n], {n, -1, z}]]  (* A193725 *)
%Y A193724 Cf. A084938, A193722, A193725.
%K A193724 nonn,tabl
%O A193724 0,4
%A A193724 _Clark Kimberling_, Aug 04 2011