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A193822 Mirror of the triangle A193821. 4
1, 1, 1, 3, 3, 2, 9, 9, 8, 4, 27, 27, 26, 20, 8, 81, 81, 80, 72, 48, 16, 243, 243, 242, 232, 192, 112, 32, 729, 729, 728, 716, 656, 496, 256, 64, 2187, 2187, 2186, 2172, 2088, 1808, 1248, 576, 128, 6561, 6561, 6560, 6544, 6432, 5984, 4864, 3072, 1280, 256 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A193822 is obtained by reversing the rows of the triangle A193821.

LINKS

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

FORMULA

Write w(n,k) for the triangle at A193821.  The triangle at A193822 is then given by w(n,n-k).

EXAMPLE

First six rows:

1

1....1

3....3....2

9....9....8.....4

27...27...26....20...8

81...81...80....72...48...16

MATHEMATICA

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}]]   (* A193821 *)

TableForm[Table[g[n], {n, -1, z}]]

Flatten[Table[g[n], {n, -1, z}]]  (* A193822 *)

CROSSREFS

Cf. A193722, A193821.

Sequence in context: A286952 A100052 A128504 * A202699 A058137 A250304

Adjacent sequences:  A193819 A193820 A193821 * A193823 A193824 A193825

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 06 2011

STATUS

approved

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Last modified June 19 07:26 EDT 2021. Contains 345126 sequences. (Running on oeis4.)