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A193727 Mirror of the triangle A193726. 3
1, 2, 1, 10, 9, 2, 50, 65, 28, 4, 250, 425, 270, 76, 8, 1250, 2625, 2200, 920, 192, 16, 6250, 15625, 16250, 9000, 2800, 464, 32, 31250, 90625, 112500, 77500, 32000, 7920, 1088, 64, 156250, 515625, 743750, 612500, 315000, 103600, 21280, 2496, 128 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A193727 is obtained by reversing the rows of the triangle A193726.

Triangle T(n,k), read by rows, given by (2,3,0,0,0,0,0,0,0,...) DELTA (1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

LINKS

Table of n, a(n) for n=0..44.

FORMULA

Write w(n,k) for the triangle at A193726.  The triangle at A193727 is then given by w(n,n-k).

T(n,k) = 2*T(n-1,k-1) + 5*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011

G.f.: (-1+3*x+x*y)/(-1+5*x+2*x*y). - R. J. Mathar, Aug 11 2015

EXAMPLE

First six rows:

     1;

     2,    1;

    10,    9,    2;

    50,   65,   28,   4;

   250,  425,  270,  76,   8;

  1250, 2625, 2200, 920, 192; 16;

MATHEMATICA

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

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}]]  (* A193726 *)

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

Flatten[Table[g[n], {n, -1, z}]]  (* A193727 *)

CROSSREFS

Cf. A084938, A193722, A193726.

Sequence in context: A127259 A152260 A286781 * A138098 A081098 A244582

Adjacent sequences:  A193724 A193725 A193726 * A193728 A193729 A193730

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 04 2011

STATUS

approved

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Last modified June 22 21:29 EDT 2021. Contains 345393 sequences. (Running on oeis4.)