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A193824 Mirror of the triangle A193823. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified February 21 06:40 EST 2019. Contains 320371 sequences. (Running on oeis4.)