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

0,2

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

Links

Table of n, a(n) for n=0..57.

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A115068.

Sequence in context: A222310 A294033 A254827 * A258631 A254947 A185810

Adjacent sequences: A193859 A193860 A193861 * A193863 A193864 A193865

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 07 2011

Status

approved