A193725 Mirror of the triangle A193724.

1, 1, 1, 3, 5, 2, 9, 21, 16, 4, 27, 81, 90, 44, 8, 81, 297, 432, 312, 112, 16, 243, 1053, 1890, 1800, 960, 272, 32, 729, 3645, 7776, 9180, 6480, 2736, 640, 64, 2187, 12393, 30618, 43092, 37800, 21168, 7392, 1472, 128, 6561, 41553, 116640, 190512, 199584, 139104, 64512, 19200, 3328, 256

Offset

0,4

Comments

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

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

Links

Table of n, a(n) for n=0..54.

Formula

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

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

G.f.: (-1+2*x+x*y)/(-1+3*x+2*x*y). - R. J. Mathar, Aug 11 2015

Example

First six rows:
   1;
   1,   1;
   3,   5,   2;
   9,  21,  16,   4;
  27,  81,  90,  44,   8;
  81, 297, 432, 312, 112, 16;

Mathematica

z = 8; a = 1; b = 2; c = 1; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193724 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193725 *)

Crossrefs

Cf. A084938, A193724.

Sequence in context: A057033 A003574 A101157 * A077952 A077975 A356378

Adjacent sequences: A193722 A193723 A193724 * A193726 A193727 A193728

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 04 2011

Status

approved