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A209133 Triangle of coefficients of polynomials u(n,x) jointly generated with A209134; see the Formula section.

%I A209133 #13 Jan 24 2020 03:26:02
%S A209133 1,2,1,2,5,4,2,9,18,10,2,13,40,56,28,2,17,70,154,176,76,2,21,108,320,
%T A209133 564,540,208,2,25,154,570,1344,1976,1640,568,2,29,208,920,2700,5304,
%U A209133 6720,4928,1552,2,33,270,1386,4848,11844,20016,22320,14688,4240
%N A209133 Triangle of coefficients of polynomials u(n,x) jointly generated with A209134; see the Formula section.
%C A209133 For a discussion and guide to related arrays, see A208510.
%C A209133 Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 3, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 10 2012
%F A209133 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209133 v(n,x) = 2x*u(n-1,x) + 2x*v(n-1,x),
%F A209133 where u(1,x)=1, v(1,x)=1.
%F A209133 From _Philippe Deléham_, Apr 10 2012: (Start)
%F A209133 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209133 G.f.: (1-2*y*x+x^2-y*x^2-2*y^2*x^2)/(1-x-2*y*x-2*y^2*x^2).
%F A209133 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A209133 First five rows:
%e A209133   1;
%e A209133   2,  1;
%e A209133   2,  5,  4;
%e A209133   2,  9, 18, 10;
%e A209133   2, 13, 40, 56, 28;
%e A209133 First three polynomials u(n,x):
%e A209133   1
%e A209133   2 + x
%e A209133   2 + 5x + 4x^2
%e A209133 From _Philippe Deléham_, Apr 10 2012: (Start)
%e A209133 (1, 1, -2, 1, 0, 0, 0, ...) DELTA (0, 1, 3, -2, 0, 0, 0, ...) begins:
%e A209133   1;
%e A209133   1,   0;
%e A209133   2,   1,   0;
%e A209133   2,   5,   4,   0;
%e A209133   2,   9,  18,  10,   0;
%e A209133   2,  13,  40,  56,  28,   0;
%e A209133   2,  17,  70, 154, 176,  76,   0;
%e A209133   2,  21, 108, 320, 564, 540, 208,  0; (End)
%t A209133 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209133 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209133 v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x];
%t A209133 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209133 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209133 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209133 TableForm[cu]
%t A209133 Flatten[%]    (* A209133 *)
%t A209133 Table[Expand[v[n, x]], {n, 1, z}]
%t A209133 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209133 TableForm[cv]
%t A209133 Flatten[%]    (* A209134 *)
%Y A209133 Cf. A209134, A208510.
%K A209133 nonn,tabl
%O A209133 1,2
%A A209133 _Clark Kimberling_, Mar 05 2012