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

0,4

Comments

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

Links

Table of n, a(n) for n=0..57.

Formula

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

Example

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

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 *)

Crossrefs

Cf. A193823.

Sequence in context: A078033 A221712 A193741 * A108075 A215120 A084145

Adjacent sequences: A193821 A193822 A193823 * A193825 A193826 A193827

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 06 2011

Status

approved