A193819 Mirror of the triangle A193818.

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

Offset

0,3

Comments

A193819 is obtained by reversing the rows of the triangle A193818.

Links

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

Formula

Write w(n,k) for the triangle at A193818. The triangle at A193819 is then given by w(n,n-k).

Triangle T(n,k), read by rows, given by (1,1,-1,1,0,0,0,0,0,0,0,...) DELTA (2,0,-2,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

T(n,k) = A153861(n,k)*2^k. - Philippe Deléham, Oct 09 2011

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1), T(0,0)=T(1,0)=1, T(1,1)=T(2,0)=2, T(2,1)=6, T(2,2)=4, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Dec 15 2013

G.f.: (1-x+x^2+2*x^2*y)/((x-1)*(-1+x+2*x*y)). - R. J. Mathar, Aug 12 2015

Example

First six rows:
  1;
  1,   2;
  2,   6,   4;
  3,  12,  16,   8;
  4,  20,  40,  40,  16;
  5,  30,  80, 120,  96,  32;

Mathematica

z = 10; c = 2; d = 1;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193818 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193819 *)

Crossrefs

Cf. A084938, A193722, A193818.

Sequence in context: A064851 A305353 A134458 * A356790 A182786 A343187

Adjacent sequences: A193816 A193817 A193818 * A193820 A193821 A193822

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 06 2011

Status

approved