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A208763 Triangle of coefficients of polynomials u(n,x) jointly generated with A208764; see the Formula section.

%I A208763 #14 Jan 24 2020 03:27:51
%S A208763 1,1,2,1,2,6,1,2,10,14,1,2,14,26,38,1,2,18,38,90,94,1,2,22,50,158,250,
%T A208763 246,1,2,26,62,242,470,762,622,1,2,30,74,342,754,1614,2138,1606,1,2,
%U A208763 34,86,458,1102,2866,4870,6170,4094,1,2,38,98,590,1514,4582
%N A208763 Triangle of coefficients of polynomials u(n,x) jointly generated with A208764; see the Formula section.
%C A208763 For a discussion and guide to related arrays, see A208510.
%C A208763 Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 19 2012
%F A208763 u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A208763 v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),
%F A208763 where u(1,x)=1, v(1,x)=1.
%F A208763 From _Philippe Deléham_, Mar 19 2012: (Start)
%F A208763 G.f.: (1-y*x+2*y*x^2-4*y^2*x^2)/(1-x-y*x+y*x^2-4*y^2*x^2).
%F A208763 T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) = 1, T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k < 0 or if k >= n. (End)
%e A208763 First five rows:
%e A208763   1;
%e A208763   1, 2;
%e A208763   1, 2,  6;
%e A208763   1, 2, 10, 14;
%e A208763   1, 2, 14, 26, 38;
%e A208763 First five polynomials u(n,x):
%e A208763   1
%e A208763   1 + 2x
%e A208763   1 + 2x +  6x^2
%e A208763   1 + 2x + 10x^2 + 14x^3
%e A208763   1 + 2x + 14x^2 + 26x^3 + 38x^4
%e A208763 From _Philippe Deléham_, Mar 19 2012: (Start)
%e A208763 (1, 0, -1, 1, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0...) begins:
%e A208763   1;
%e A208763   1,  0;
%e A208763   1,  2,  0;
%e A208763   1,  2,  6,  0;
%e A208763   1,  2, 10, 14,  0;
%e A208763   1,  2, 14, 26, 38,  0;
%e A208763   1,  2, 18, 38, 90, 94,  0; (End)
%t A208763 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208763 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208763 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x];
%t A208763 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208763 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208763 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208763 TableForm[cu]
%t A208763 Flatten[%]    (* A208763 *)
%t A208763 Table[Expand[v[n, x]], {n, 1, z}]
%t A208763 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208763 TableForm[cv]
%t A208763 Flatten[%]    (* A208764 *)
%Y A208763 Cf. A208764, A208510.
%K A208763 nonn,tabl
%O A208763 1,3
%A A208763 _Clark Kimberling_, Mar 02 2012