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A193725 Mirror of the triangle A193724.

%I A193725 #24 Jan 22 2020 20:11:50
%S A193725 1,1,1,3,5,2,9,21,16,4,27,81,90,44,8,81,297,432,312,112,16,243,1053,
%T A193725 1890,1800,960,272,32,729,3645,7776,9180,6480,2736,640,64,2187,12393,
%U A193725 30618,43092,37800,21168,7392,1472,128,6561,41553,116640,190512,199584,139104,64512,19200,3328,256
%N A193725 Mirror of the triangle A193724.
%C A193725 A193725 is obtained by reversing the rows of the triangle A193724.
%C A193725 Triangle T(n,k), read by rows, given by [1,2,0,0,0,0,...] DELTA [1,1,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 04 2011
%F A193725 Write w(n,k) for the triangle at A193724.  The triangle at A193725 is then given by w(n,n-k).
%F A193725 T(n,k) = 2*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=T(1,1)=1. - _Philippe Deléham_, Oct 05 2011
%F A193725 G.f.: (-1+2*x+x*y)/(-1+3*x+2*x*y). - _R. J. Mathar_, Aug 11 2015
%e A193725 First six rows:
%e A193725    1;
%e A193725    1,   1;
%e A193725    3,   5,   2;
%e A193725    9,  21,  16,   4;
%e A193725   27,  81,  90,  44,   8;
%e A193725   81, 297, 432, 312, 112, 16;
%t A193725 z = 8; a = 1; b = 2; c = 1; d = 1;
%t A193725 p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t A193725 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193725 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193725 g[n_] := CoefficientList[w[n, x], {x}]
%t A193725 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193725 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193724 *)
%t A193725 TableForm[Table[g[n], {n, -1, z}]]
%t A193725 Flatten[Table[g[n], {n, -1, z}]]  (* A193725 *)
%Y A193725 Cf. A084938, A193724.
%K A193725 nonn,tabl
%O A193725 0,4
%A A193725 _Clark Kimberling_, Aug 04 2011