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A208755 Triangle of coefficients of polynomials u(n,x) jointly generated with A208756; see the Formula section.

%I A208755 #19 Jan 22 2020 20:13:23
%S A208755 1,1,2,1,2,4,1,2,6,8,1,2,8,14,16,1,2,10,20,34,32,1,2,12,26,56,78,64,1,
%T A208755 2,14,32,82,140,178,128,1,2,16,38,112,218,352,398,256,1,2,18,44,146,
%U A208755 312,594,852,882,512,1,2,20,50,184,422,912,1530,2040,1934,1024
%N A208755 Triangle of coefficients of polynomials u(n,x) jointly generated with A208756; see the Formula section.
%C A208755 For a discussion and guide to related arrays, see A208510.
%C A208755 Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 04 2012
%F A208755 u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A208755 v(n,x) = x*u(n-1,x) + x*v(n-1,x),
%F A208755 where u(1,x)=1, v(1,x)=1.
%F A208755 From _Philippe Deléham_, Mar 04 2012: (Start)
%F A208755 T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 1, T(2,1) = 1 and T(n,k) = 0 if k < 0 or if k > n. (End)
%F A208755 G.f.: -(1+x*y)*x*y/(-1+x*y-x^2*y+2*x^2*y^2+x). - _R. J. Mathar_, Aug 11 2015
%e A208755 First five rows:
%e A208755   1;
%e A208755   1, 2;
%e A208755   1, 2, 4;
%e A208755   1, 2, 6,  8;
%e A208755   1, 2, 8, 14, 16;
%e A208755 First five polynomials u(n,x):
%e A208755   1
%e A208755   1 + 2x
%e A208755   1 + 2x + 4x^2
%e A208755   1 + 2x + 6x^2 +  8x^3
%e A208755   1 + 2x + 8x^2 + 14x^3 + 16x^4
%e A208755 From _Philippe Deléham_, Mar 04 2012: (Start)
%e A208755 Triangle (1, 0, -1, 1, 0, 0, 0...) DELTA (0, 2, 0, -1, 0, 0, 0, ...) begins:
%e A208755   1;
%e A208755   1,  0;
%e A208755   1,  2,  0;
%e A208755   1,  2,  4,  0;
%e A208755   1,  2,  6,  8,  0;
%e A208755   1,  2,  8, 14, 16,  0;
%e A208755   1,  2, 10, 20, 34, 32,  0; (End)
%t A208755 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208755 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208755 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
%t A208755 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208755 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208755 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208755 TableForm[cu]
%t A208755 Flatten[%]    (* A208755 *)
%t A208755 Table[Expand[v[n, x]], {n, 1, z}]
%t A208755 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208755 TableForm[cv]
%t A208755 Flatten[%]    (* A208756 *)
%Y A208755 Cf. A208756, A208510.
%K A208755 nonn,tabl
%O A208755 1,3
%A A208755 _Clark Kimberling_, Mar 01 2012