A193857 Mirror of the triangle A193856.

1, 5, 1, 19, 8, 1, 65, 43, 11, 1, 211, 194, 76, 14, 1, 665, 793, 422, 118, 17, 1, 2059, 3044, 2059, 776, 169, 20, 1, 6305, 11191, 9221, 4387, 1283, 229, 23, 1, 19171, 39878, 38854, 22382, 8236, 1970, 298, 26, 1, 58025, 138805, 156440, 106000, 47090

Offset: 0

Author

Clark Kimberling, Aug 07 2011

Keywords

• nonn
• tabl

Comments

A193857 is obtained by reversing the rows of the triangle A193856.

Examples

			First six rows:
1
5.....1
19....8.....1
65....43....11....1
211...194...76....14....1
665...793...422...118...17...1
		

Crossrefs

Cf. A193856.

Programs

Mathematica

z = 10;
p[n_, x_] := (2 x + 1)^n;
q[n_, x_] := (x + 1)^n;
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}]]  (* A193856 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193857 *)

Formula

Write w(n,k) for the triangle at A193856. The triangle at A193857 is then given by w(n,n-k).