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A208758 Triangle of coefficients of polynomials v(n,x) jointly generated with A208757; see the Formula section.

%I A208758 #15 Jan 24 2020 03:28:20
%S A208758 1,0,3,0,1,8,0,1,4,22,0,1,4,16,60,0,1,4,18,56,164,0,1,4,20,68,188,448,
%T A208758 0,1,4,22,80,248,608,1224,0,1,4,24,92,312,864,1920,3344,0,1,4,26,104,
%U A208758 380,1152,2928,5952,9136,0,1,4,28,116,452,1472,4128,9696,18192
%N A208758 Triangle of coefficients of polynomials v(n,x) jointly generated with A208757; see the Formula section.
%C A208758 For a discussion and guide to related arrays, see A208510.
%C A208758 As triangle T(n,k) with 0 <= k <= n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 02 2012
%F A208758 u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A208758 v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
%F A208758 where u(1,x)=1, v(1,x)=1.
%F A208758 As triangle T(n,k), 0 <= k <= n: g.f.: (1-x-y*x)/(1-(1+2*y)*x -2*y(y-1)*x^2). - _Philippe Deléham_, Mar 02 2012
%F A208758 As triangle T(n,k), 0 <= k <= n: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3 and T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Mar 02 2012
%e A208758 First five rows:
%e A208758   1;
%e A208758   0,  3;
%e A208758   0,  1,  8;
%e A208758   0,  1,  4, 22;
%e A208758   0,  1,  4, 16, 60;
%e A208758 First five polynomials v(n,x):
%e A208758   1
%e A208758      3x
%e A208758       x + 8x^2
%e A208758       x + 4x^2 + 22x^3
%e A208758       x + 4x^2 + 16x^3 + 60^x4
%t A208758 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208758 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208758 v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
%t A208758 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208758 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208758 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208758 TableForm[cu]
%t A208758 Flatten[%]  (* A208757 *)
%t A208758 Table[Expand[v[n, x]], {n, 1, z}]
%t A208758 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208758 TableForm[cv]
%t A208758 Flatten[%]  (* A208758 *)
%Y A208758 Cf. A208757, A208510.
%K A208758 nonn,tabl
%O A208758 1,3
%A A208758 _Clark Kimberling_, Mar 02 2012