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A193729 Mirror of the triangle A193728. 3
1, 1, 2, 3, 10, 8, 9, 42, 64, 32, 27, 162, 360, 352, 128, 81, 594, 1728, 2496, 1792, 512, 243, 2106, 7560, 14400, 15360, 8704, 2048, 729, 7290, 31104, 73440, 103680, 87552, 40960, 8192, 2187, 24786, 122472, 344736, 604800, 677376, 473088, 188416 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A193729 is obtained by reversing the rows of the triangle A193728.

Triangle T(n,k), read by rows, given by [1,2,0,0,0,0,...] DELTA [2,2,0,0,0,0,...] where DELTA is the operator defined in A084938. - From DELEHA Philippe, Oct 05 2011.

LINKS

Table of n, a(n) for n=0..43.

FORMULA

Write w(n,k) for the triangle at A193728.  The triangle at A193729 is then given by w(n,n-k).

T(n,k)=4*T(n-1,k-1)+3*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2 - From Philippe Deléham, Oct 05 2011.

G.f.: (-1+2*x+2*x*y)/(-1+3*x+4*x*y). - R. J. Mathar, Aug 11 2015

EXAMPLE

First six rows:

1

1....2

3....10....8

9....42....64....32

27...162...360...352...128

81...594...1728..2496..1792..512

MATHEMATICA

z = 8; a = 1; b = 2; c = 2; d = 1;

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}]]  (* A193728 *)

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

Flatten[Table[g[n], {n, -1, z}]]   (* A193729 *)

CROSSREFS

Cf. A084938, A193722, A193728.

Sequence in context: A123167 A141670 A278561 * A303115 A074068 A283436

Adjacent sequences:  A193726 A193727 A193728 * A193730 A193731 A193732

KEYWORD

nonn,tabl,changed

AUTHOR

Clark Kimberling, Aug 04 2011

STATUS

approved

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Last modified November 21 12:11 EST 2019. Contains 329370 sequences. (Running on oeis4.)