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A208333 Triangle of coefficients of polynomials v(n,x) jointly generated with A208332; see the Formula section.

%I A208333 #19 Jan 24 2020 03:27:10
%S A208333 1,0,4,0,2,10,0,2,6,28,0,2,6,24,76,0,2,6,28,80,208,0,2,6,32,100,264,
%T A208333 568,0,2,6,36,120,360,840,1552,0,2,6,40,140,464,1232,2624,4240,0,2,6,
%U A208333 44,160,576,1680,4128,8064,11584,0,2,6,48,180,696,2184,5952
%N A208333 Triangle of coefficients of polynomials v(n,x) jointly generated with A208332; see the Formula section.
%C A208333 As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 28 2012
%F A208333 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A208333 v(n,x) = 2x*u(n-1,x) + 2x*v(n-1,x),
%F A208333 where u(1,x)=1, v(1,x)=1.
%F A208333 From _Philippe Deléham_, Feb 28 2012: (Start)
%F A208333 As triangle with 0 <= k <= n:
%F A208333 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) = 4 and T(n,k) = 0 if k < 0 or if k > n.
%F A208333 G.f.: (1-x+2*y*x)/(1-x-2*y*x+2*y*x^2-2*y^2*x^2).
%F A208333 T(n,n) = A026150(n+1).
%F A208333 Sum_{k=0..n} T(n,k) = A003946(n). (End)
%e A208333 First five rows:
%e A208333   1;
%e A208333   0,  4;
%e A208333   0,  2, 10;
%e A208333   0,  2,  6, 28;
%e A208333   0,  2,  6, 24, 76;
%e A208333 First five polynomials u(n,x):
%e A208333   1
%e A208333       4x
%e A208333       2x + 10x^2
%e A208333       2x +  6x^2 + 28x^3
%e A208333       2x +  6x^2 + 24x^3 + 76x^4.
%t A208333 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208333 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208333 v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x];
%t A208333 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208333 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208333 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208333 TableForm[cu]
%t A208333 Flatten[%]  (* A208332 *)
%t A208333 Table[Expand[v[n, x]], {n, 1, z}]
%t A208333 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208333 TableForm[cv]
%t A208333 Flatten[%]  (* A208333 *)
%Y A208333 Cf. A003946, A026150, A208332.
%K A208333 nonn,tabl
%O A208333 1,3
%A A208333 _Clark Kimberling_, Feb 26 2012