A193725 - OEIS

%I #24 Jan 22 2020 20:11:50

%S 1,1,1,3,5,2,9,21,16,4,27,81,90,44,8,81,297,432,312,112,16,243,1053,

%T 1890,1800,960,272,32,729,3645,7776,9180,6480,2736,640,64,2187,12393,

%U 30618,43092,37800,21168,7392,1472,128,6561,41553,116640,190512,199584,139104,64512,19200,3328,256

%N Mirror of the triangle A193724.

%C A193725 is obtained by reversing the rows of the triangle A193724.

%C 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 Write w(n,k) for the triangle at A193724.  The triangle at A193725 is then given by w(n,n-k).

%F 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 G.f.: (-1+2*x+x*y)/(-1+3*x+2*x*y). - _R. J. Mathar_, Aug 11 2015

%e First six rows:

%e    1;

%e    1,   1;

%e    3,   5,   2;

%e    9,  21,  16,   4;

%e   27,  81,  90,  44,   8;

%e   81, 297, 432, 312, 112, 16;

%t z = 8; a = 1; b = 2; c = 1; d = 1;

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

%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}]] (* A193724 *)

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

%t Flatten[Table[g[n], {n, -1, z}]]  (* A193725 *)

%Y Cf. A084938, A193724.

%K nonn,tabl

%O 0,4

%A _Clark Kimberling_, Aug 04 2011