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A193819 Mirror of the triangle A193818.

%I A193819 #21 Jan 26 2020 01:01:39
%S A193819 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 A193819 336,224,64,7,56,224,560,896,896,512,128,8,72,336,1008,2016,2688,2304,
%U A193819 1152,256,9,90,480,1680,4032,6720,7680,5760,2560,512,10,110,660
%N A193819 Mirror of the triangle A193818.
%C A193819 A193819 is obtained by reversing the rows of the triangle A193818.
%F A193819 Write w(n,k) for the triangle at A193818.  The triangle at A193819 is then given by w(n,n-k).
%F A193819 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 A193819 T(n,k) = A153861(n,k)*2^k. - _Philippe Deléham_, Oct 09 2011
%F A193819 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 A193819 G.f.: (1-x+x^2+2*x^2*y)/((x-1)*(-1+x+2*x*y)). - _R. J. Mathar_, Aug 12 2015
%e A193819 First six rows:
%e A193819   1;
%e A193819   1,   2;
%e A193819   2,   6,   4;
%e A193819   3,  12,  16,   8;
%e A193819   4,  20,  40,  40,  16;
%e A193819   5,  30,  80, 120,  96,  32;
%t A193819 z = 10; c = 2; d = 1;
%t A193819 p[0, x_] := 1
%t A193819 p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
%t A193819 q[n_, x_] := (c*x + d)^n
%t A193819 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193819 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193819 g[n_] := CoefficientList[w[n, x], {x}]
%t A193819 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193819 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193818 *)
%t A193819 TableForm[Table[g[n], {n, -1, z}]]
%t A193819 Flatten[Table[g[n], {n, -1, z}]]   (* A193819 *)
%Y A193819 Cf. A084938, A193722, A193818.
%K A193819 nonn,tabl
%O A193819 0,3
%A A193819 _Clark Kimberling_, Aug 06 2011