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A193817 Mirror of the triangle A193816. 2
1, 2, 1, 6, 5, 1, 14, 17, 7, 1, 30, 49, 31, 9, 1, 62, 129, 111, 49, 11, 1, 126, 321, 351, 209, 71, 13, 1, 254, 769, 1023, 769, 351, 97, 15, 1, 510, 1793, 2815, 2561, 1471, 545, 127, 17, 1, 1022, 4097, 7423, 7937, 5503, 2561, 799, 161, 19, 1, 2046, 9217, 18943 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A193817 is obtained by reversing the rows of the triangle A193816.

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

LINKS

Table of n, a(n) for n=0..57.

FORMULA

Write w(n,k) for the triangle at A193816.  The triangle at A193817 is then given by w(n,n-k).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(2,0)=6, T(2,1)=5, T(2,2)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Dec 15 2013

G.f.: (1-x+2*x^2+x^2*y)/((x-1)*(-1+2*x+x*y)). - R. J. Mathar, Aug 12 2015

EXAMPLE

First six rows:

   1;

   2,   1;

   6,   5,   1;

  14,  17,   7,   1;

  30,  49,  31,   9,   1;

  62, 129, 111,  49,  11,   1;

MATHEMATICA

z = 10; c = 1; d = 2;

p[0, x_] := 1

p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;

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}]]  (* A193816 *)

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

Flatten[Table[g[n], {n, -1, z}]]   (* A193817 *)

CROSSREFS

Cf. A084938, A193722, A193816.

Sequence in context: A259332 A108767 A046817 * A227159 A294439 A008970

Adjacent sequences:  A193814 A193815 A193816 * A193818 A193819 A193820

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 06 2011

STATUS

approved

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Last modified August 9 05:12 EDT 2020. Contains 336319 sequences. (Running on oeis4.)