|
%I #23 Jan 27 2020 01:34:03
%S 1,2,1,6,5,1,18,21,8,1,54,81,45,11,1,162,297,216,78,14,1,486,1053,945,
%T 450,120,17,1,1458,3645,3888,2295,810,171,20,1,4374,12393,15309,10773,
%U 4725,1323,231,23,1,13122,41553,58320,47628,24948,8694,2016,300,26,1
%N Mirror of the fusion triangle A193722.
%C A193723 is obtained by reversing the rows of the triangle A193722.
%C Triangle T(n,k), read by rows, given by [2,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 04 2011
%C From _Philippe Deléham_, Nov 14 2011: (Start)
%C Riordan array ((1-x)/(1-3x), x/(1-3x)).
%C Product A200139*A007318 as infinite lower triangular arrays. (End)
%F Write w(n,k) for the triangle at A193722. The triangle at A193723 is then given by w(n,n-k).
%F T(n,k) = T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - _Philippe Deléham_, Oct 05 2011
%F From _Philippe Deléham_, Nov 14 2011: (Start)
%F Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A052268(n), A055276(n), A196731(n) for x=-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.
%F T(n,k) = Sum_{j>=0} T(n-1-j,k-1)*3^j.
%F G.f.: (1-x)/(1-(3+y)*x).
%e First six rows:
%e 1;
%e 2, 1;
%e 6, 5, 1;
%e 18, 21, 8, 1;
%e 54, 81, 45, 11, 1;
%e 162, 297, 216, 78, 14, 1;
%t z = 9; a = 1; b = 1; c = 1; d = 2;
%t p[n_, x_] := (a*x + b)^n ; 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}]] (* A193722 *)
%t TableForm[Table[g[n], {n, -1, z}]]
%t Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)
%Y Cf. A084938, A193722, A052924 (antidiagonal sums), Diagonals: A000012, A016789, A081266, Columns: A025192, A081038.
%K nonn,tabl
%O 0,2
%A _Clark Kimberling_, Aug 04 2011
|
