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%I #18 Jan 26 2020 01:01:34
%S 1,2,1,6,5,1,14,17,7,1,30,49,31,9,1,62,129,111,49,11,1,126,321,351,
%T 209,71,13,1,254,769,1023,769,351,97,15,1,510,1793,2815,2561,1471,545,
%U 127,17,1,1022,4097,7423,7937,5503,2561,799,161,19,1,2046,9217,18943
%N Mirror of the triangle A193816.
%C A193817 is obtained by reversing the rows of the triangle A193816.
%C Triangle T(n,k), read by rows, given by (2,1,-2,2,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined by A084938. - _Philippe Deléham_, Oct 05 2011
%F Write w(n,k) for the triangle at A193816. The triangle at A193817 is then given by w(n,n-k).
%F T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(2,0)=6, T(2,1)=5, T(2,2)=1, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Dec 15 2013
%F G.f.: (1-x+2*x^2+x^2*y)/((x-1)*(-1+2*x+x*y)). - _R. J. Mathar_, Aug 12 2015
%e First six rows:
%e 1;
%e 2, 1;
%e 6, 5, 1;
%e 14, 17, 7, 1;
%e 30, 49, 31, 9, 1;
%e 62, 129, 111, 49, 11, 1;
%t z = 10; c = 1; d = 2;
%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}]] (* A193816 *)
%t TableForm[Table[g[n], {n, -1, z}]]
%t Flatten[Table[g[n], {n, -1, z}]] (* A193817 *)
%Y Cf. A084938, A193722, A193816.
%K nonn,tabl
%O 0,2
%A _Clark Kimberling_, Aug 06 2011
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