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A193731 Mirror of the triangle A193730. 3
1, 1, 2, 3, 8, 4, 9, 30, 28, 8, 27, 108, 144, 80, 16, 81, 378, 648, 528, 208, 32, 243, 1296, 2700, 2880, 1680, 512, 64, 729, 4374, 10692, 14040, 10800, 4896, 1216, 128, 2187, 14580, 40824, 63504, 60480, 36288, 13440, 2816, 256, 6561, 48114, 151632 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A193731 is obtained by reversing the rows of the triangle A193730.

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

LINKS

Table of n, a(n) for n=0..47.

FORMULA

Write w(n,k) for the triangle at A193730.  The triangle at A193731 is then given by w(n,n-k).

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

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

EXAMPLE

First six rows:

   1;

   1,   2;

   3,   8,   4;

   9,  30,  28,   8;

  27, 108, 144,  80,  16;

  81, 378, 648, 528, 208, 32;

MATHEMATICA

z = 8; a = 2; b = 1; 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}]]  (* A193730 *)

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

Flatten[Table[g[n], {n, -1, z}]]     (* A193731 *)

CROSSREFS

Cf. A084938, A193722, A193730.

Sequence in context: A110142 A158928 A198369 * A193975 A224665 A098514

Adjacent sequences:  A193728 A193729 A193730 * A193732 A193733 A193734

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 04 2011

STATUS

approved

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Last modified October 20 08:28 EDT 2021. Contains 348099 sequences. (Running on oeis4.)