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

Author

Clark Kimberling, Aug 06 2011

Keywords

• nonn
• tabl

Comments

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

Examples

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

Crossrefs

Cf. A084938, A193722, A193818.

Programs

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 *)

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