A193824 Mirror of the triangle A193823.

1, 1, 1, 3, 3, 1, 9, 9, 5, 1, 27, 27, 19, 7, 1, 81, 81, 65, 33, 9, 1, 243, 243, 211, 131, 51, 11, 1, 729, 729, 665, 473, 233, 73, 13, 1, 2187, 2187, 2059, 1611, 939, 379, 99, 15, 1, 6561, 6561, 6305, 5281, 3489, 1697, 577, 129, 17, 1, 19683, 19683, 19171

Offset: 0

Author

Clark Kimberling, Aug 06 2011

Keywords

• nonn
• tabl

Comments

A193824 is obtained by reversing the rows of the triangle A193823.

Examples

			First six rows:
1
1....1
3....3....1
9....9....5.....1
27...27...19....7...1
81...81...65....33...9...1
		

Crossrefs

Cf. A193823.

Programs

Mathematica

z = 10; a = 2; b = 1;
p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
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}]]   (* A193823 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193824 *)

Formula

Write w(n,k) for the triangle at A193823. The triangle at A193824 is then given by w(n,n-k).