A193861 Mirror of the triangle A193860.

1, 5, 1, 19, 7, 1, 65, 33, 9, 1, 211, 131, 51, 11, 1, 665, 473, 233, 73, 13, 1, 2059, 1611, 939, 379, 99, 15, 1, 6305, 5281, 3489, 1697, 577, 129, 17, 1, 19171, 16867, 12259, 6883, 2851, 835, 163, 19, 1, 58025, 52905, 41385, 26025, 12585, 4521, 1161

Offset: 0

Author

Clark Kimberling, Aug 07 2011

Keywords

• nonn
• tabl

Comments

A193861 is obtained by reversing the rows of the triangle A193860.

Examples

			First six rows:
1
5.....1
19....7.....1
65....33....9.....1
211...131...51....11...1
665...473...233...73...13...1
		

Crossrefs

Cf. A193860.

Programs

Mathematica

z = 10;
p[n_, x_] := (2 x + 1)^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}]]  (* A193860 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193861  *)

Formula

Write w(n,k) for the triangle at A193860. The triangle at A193861 is then given by w(n,n-k).