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A209128 Triangle of coefficients of polynomials u(n,x) jointly generated with A209129; see the Formula section.

%I A209128 #14 Jan 24 2020 03:27:27
%S A209128 1,2,1,2,4,3,2,6,12,7,2,8,22,32,17,2,10,34,70,86,41,2,12,48,124,216,
%T A209128 228,99,2,14,64,196,428,644,600,239,2,16,82,288,744,1408,1876,1568,
%U A209128 577,2,18,102,402,1188,2664,4476,5364,4074,1393,2,20,124,540,1786
%N A209128 Triangle of coefficients of polynomials u(n,x) jointly generated with A209129; see the Formula section.
%C A209128 For a discussion and guide to related arrays, see A208510.
%C A209128 Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 21 2012
%C A209128 Row sums are powers of 3 (A000244). - _Philippe Deléham_, Mar 21 2012
%F A209128 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209128 v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
%F A209128 where u(1,x)=1, v(1,x)=1.
%F A209128 From _Philippe Deléham_, Mar 21 2012: (Start)
%F A209128 As DELTA-triangle with 0 <= k <= n:
%F A209128 G.f.: (1-2*y*x+x^2-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).
%F A209128 T(n,k) = T(n-1,k-1) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A209128 First five rows:
%e A209128   1;
%e A209128   2,  1;
%e A209128   2,  4,  3;
%e A209128   2,  6, 12,  7;
%e A209128   2,  8, 22, 32, 17;
%e A209128 First three polynomials u(n,x):
%e A209128   1
%e A209128   2 + x
%e A209128   2 + 4x + 3x^2
%e A209128 From _Philippe Deléham_, Mar 21 2012: (Start)
%e A209128 (1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
%e A209128   1;
%e A209128   1,  0;
%e A209128   2,  1,  0;
%e A209128   2,  4,  3,  0;
%e A209128   2,  6, 12,  7,  0;
%e A209128   2,  8, 22, 32, 17,  0; (End)
%t A209128 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209128 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209128 v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
%t A209128 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209128 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209128 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209128 TableForm[cu]
%t A209128 Flatten[%]    (* A209128 *)
%t A209128 Table[Expand[v[n, x]], {n, 1, z}]
%t A209128 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209128 TableForm[cv]
%t A209128 Flatten[%]    (* A209129 *)
%Y A209128 Cf. A209129, A208510.
%K A209128 nonn,tabl
%O A209128 1,2
%A A209128 _Clark Kimberling_, Mar 05 2012