A193723 Mirror of the fusion triangle A193722.

1, 2, 1, 6, 5, 1, 18, 21, 8, 1, 54, 81, 45, 11, 1, 162, 297, 216, 78, 14, 1, 486, 1053, 945, 450, 120, 17, 1, 1458, 3645, 3888, 2295, 810, 171, 20, 1, 4374, 12393, 15309, 10773, 4725, 1323, 231, 23, 1, 13122, 41553, 58320, 47628, 24948, 8694, 2016, 300, 26, 1

Offset

0,2

Comments

A193723 is obtained by reversing the rows of the triangle A193722.

Triangle T(n,k), read by rows, given by [2,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011

From Philippe Deléham, Nov 14 2011: (Start)

Riordan array ((1-x)/(1-3x), x/(1-3x)).

Product A200139*A007318 as infinite lower triangular arrays. (End)

Links

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

Formula

Write w(n,k) for the triangle at A193722. The triangle at A193723 is then given by w(n,n-k).

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011

From Philippe Deléham, Nov 14 2011: (Start)

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A052268(n), A055276(n), A196731(n) for x=-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.

T(n,k) = Sum_{j>=0} T(n-1-j,k-1)*3^j.

G.f.: (1-x)/(1-(3+y)*x). (End)

Example

First six rows:
    1;
    2,   1;
    6,   5,   1;
   18,  21,   8,   1;
   54,  81,  45,  11,   1;
  162, 297, 216,  78,  14,   1;

Mathematica

z = 9; a = 1; b = 1; c = 1; d = 2;
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}]] (* A193722 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)

Crossrefs

Cf. A084938, A193722, A052924 (antidiagonal sums), Diagonals: A000012, A016789, A081266, Columns: A025192, A081038.

Sequence in context: A294439 A008970 A055896 * A260914 A159965 A116395

Adjacent sequences: A193720 A193721 A193722 * A193724 A193725 A193726

Keyword

nonn,tabl,changed

Author

Clark Kimberling, Aug 04 2011

Status

approved