A193862 Mirror of the triangle A115068.

1, 2, 2, 3, 6, 4, 4, 12, 16, 8, 5, 20, 40, 40, 16, 6, 30, 80, 120, 96, 32, 7, 42, 140, 280, 336, 224, 64, 8, 56, 224, 560, 896, 896, 512, 128, 9, 72, 336, 1008, 2016, 2688, 2304, 1152, 256, 10, 90, 480, 1680, 4032, 6720, 7680, 5760, 2560, 512, 11, 110, 660

Offset: 0

Author

Clark Kimberling, Aug 07 2011

Keywords

• nonn
• tabl

Comments

A193862 is obtained by reversing the rows of the triangle A115068.

Riordan array (1/(1-x)^2, 2*x/(1-x)). - Philippe Deléham, Jan 29 2014

Let P(n, x) := Sum_{k=1..n} T(n, k)*x^k. Then P(n, P(m, x)) = P(n*m, x) for all n and m in Z. - Michael Somos, Apr 10 2020

Examples

			First six rows:
1
2...2
3...6....4
4...12...16...8
5...20...40...40....16
6...30...80...120...96...32
Production matrix begins
2......2
-1/2...1...2
1/4....0...1...2
-1/8...0...0...1...2
1/16...0...0...0...1...2
-1/32..0...0...0...0...1...2
1/64...0...0...0...0...0...1...2
-1/128.0...0...0...0...0...0...1...2
1/256..0...0...0...0...0...0...0...1...2
- _Philippe Deléham_, Jan 29 2014
		

Crossrefs

Cf. A115068.

Programs

Mathematica

z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + 1;
q[n_, x_] := (2 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}]]  (* A115068 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193862 *)
T[ n_, k_] := Binomial[n, k]/2 2^k; (* Michael Somos, Apr 10 2020 *)

PARI

{T(n, k) = binomial(n, k)/2 * 2^k}; /* Michael Somos, Apr 10 2020 */

Formula

Write w(n,k) for the triangle at A115068. The triangle at A193862 is then given by w(n,n-k).

T(n, k) = binomial(n, k)/2 * 2^k. - Michael Somos, Apr 10 2020