A193980 Mirror of the triangle A193979.

1, 3, 2, 9, 5, 3, 21, 13, 7, 4, 41, 28, 17, 9, 5, 71, 52, 35, 21, 11, 6, 113, 87, 63, 42, 25, 13, 7, 169, 135, 103, 74, 49, 29, 15, 8, 241, 198, 157, 119, 85, 56, 33, 17, 9, 331, 278, 227, 179, 135, 96, 63, 37, 19, 10, 441, 377, 315, 256, 201, 151, 107, 70, 41, 21, 11

Offset: 0

Author

Clark Kimberling, Aug 10 2011

Keywords

• nonn
• tabl

Comments

A193980 is obtained by reversing the rows of the triangle A193977.

Examples

			First six rows:
1
3....2
9....5....3
21...13...7....4
41...28...17...9....5
71...52...35...21...11...6
		

Crossrefs

Cf. A193979.

Programs

Mathematica

z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n;
q[n_, x_] := p[n, x];
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}]]  (* A193979 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A193980 *)

Formula

Write w(n,k) for the triangle at A193979. The triangle at A193980 is then given by w(n,n-k).