A193851 Mirror of the triangle A193850.

2, 8, 4, 26, 20, 8, 80, 72, 48, 16, 242, 232, 192, 112, 32, 728, 716, 656, 496, 256, 64, 2186, 2172, 2088, 1808, 1248, 576, 128, 6560, 6544, 6432, 5984, 4864, 3072, 1280, 256, 19682, 19664, 19520, 18848, 16832, 12800, 7424, 2816, 512, 59048, 59028

Offset: 0

Author

Clark Kimberling, Aug 07 2011

Keywords

• nonn
• tabl

Comments

A193851 is obtained by reversing the rows of the triangle A193850.

Examples

			First six rows:
2
8....4
26...20....8
80...72...40..16
242...232...192...112...32
728...716...656...496..256..64
		

Crossrefs

Cf. A193850, A193853.

Programs

Mathematica

z = 10;
p[n_, x_] := (x + 2)^n;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193850 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193851  *)
TableForm[Table[Reverse[h[n]/2], {n, 0, z}]]
Flatten[Table[Reverse[h[n]]/2, {n, -1, z}]] (* A193852 *)
TableForm[Table[h[n]/2, {n, 0, z}]]
Flatten[Table[h[n]/2, {n, -1, z}]]  (* A193853 *)

Formula

Write w(n,k) for the triangle at A193850. The triangle at A193851 is then given by w(n,n-k).