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Mirror of the triangle A193818.
2

%I #21 Jan 26 2020 01:01:39

%S 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,

%T 336,224,64,7,56,224,560,896,896,512,128,8,72,336,1008,2016,2688,2304,

%U 1152,256,9,90,480,1680,4032,6720,7680,5760,2560,512,10,110,660

%N Mirror of the triangle A193818.

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

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

%F 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

%F T(n,k) = A153861(n,k)*2^k. - _Philippe Deléham_, Oct 09 2011

%F 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

%F G.f.: (1-x+x^2+2*x^2*y)/((x-1)*(-1+x+2*x*y)). - _R. J. Mathar_, Aug 12 2015

%e First six rows:

%e 1;

%e 1, 2;

%e 2, 6, 4;

%e 3, 12, 16, 8;

%e 4, 20, 40, 40, 16;

%e 5, 30, 80, 120, 96, 32;

%t z = 10; c = 2; d = 1;

%t p[0, x_] := 1

%t p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;

%t q[n_, x_] := (c*x + d)^n

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193818 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A193819 *)

%Y Cf. A084938, A193722, A193818.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Aug 06 2011