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

%I A208764 #11 Sep 08 2013 19:59:31
%S A208764 1,0,3,0,2,7,0,2,6,19,0,2,6,26,47,0,2,6,34,78,123,0,2,6,42,110,258,
%T A208764 311,0,2,6,50,142,426,758,803,0,2,6,58,174,626,1366,2282,2047,0,2,6,
%U A208764 66,206,858,2134,4594,6558,5259,0,2,6,74,238,1122,3062,7866,14334
%N A208764 Triangle of coefficients of polynomials v(n,x) jointly generated with A208763; see the Formula section.
%C A208764 For a discussion and guide to related arrays, see A208510.
%C A208764 As triangle T(n,k) with 0<=k<=n, it is (0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -4/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 02 2012
%F A208764 u(n,x)=u(n-1,x)+2x*v(n-1,x),
%F A208764 v(n,x)=2x*u(n-1,x)+x*v(n-1,x),
%F A208764 where u(1,x)=1, v(1,x)=1.
%F A208764 As triangle T(n,k), 0 <=k<=n :
%F A208764 G.f.: (1-x+2y*x)/(1-(1+y)*x -(4*y^2-y)*x^2). - _Philippe Deléham_, Mar 02 2012
%F A208764 As triangle T(n,k), 0<=k<=n : T(n,k) = T(n-1,k) + T(n-1,k-1) + 4*T(n-2,k-2) - T(n-2,k-1) 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 A208764 First five rows:
%e A208764 1
%e A208764 0...3
%e A208764 0...2...7
%e A208764 0...2...6...19
%e A208764 0...2...6...26...47
%e A208764 First five polynomials v(n,x):
%e A208764 1
%e A208764 3x
%e A208764 2x + 7x^2
%e A208764 2x + 6x^2 + 19x^3
%e A208764 2x + 6x^2 + 26x^3 + 47x^4
%t A208764 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208764 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208764 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x];
%t A208764 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208764 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208764 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208764 TableForm[cu]
%t A208764 Flatten[%]    (* A208763 *)
%t A208764 Table[Expand[v[n, x]], {n, 1, z}]
%t A208764 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208764 TableForm[cv]
%t A208764 Flatten[%]    (* A208764 *)
%Y A208764 Cf. A208763, A208510.
%K A208764 nonn,tabl
%O A208764 1,3
%A A208764 _Clark Kimberling_, Mar 02 2012