A193817 Mirror of the triangle A193816.

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

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