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A208748 Triangle of coefficients of polynomials v(n,x) jointly generated with A208747; see the Formula section.

%I A208748 #17 Aug 11 2015 13:56:35
%S A208748 1,0,4,0,2,12,0,2,8,40,0,2,8,40,128,0,2,8,48,160,416,0,2,8,56,208,640,
%T A208748 1344,0,2,8,64,256,928,2432,4352,0,2,8,72,304,1248,3840,9088,14080,0,
%U A208748 2,8,80,352,1600,5504,15616,33280,45568,0,2,8,88,400,1984,7424
%N A208748 Triangle of coefficients of polynomials v(n,x) jointly generated with A208747; see the Formula section.
%C A208748 For a discussion and guide to related arrays, see A208510.
%C A208748 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, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 14 2012
%F A208748 u(n,x)=u(n-1,x)+2x*v(n-1,x),
%F A208748 v(n,x)=2x*u(n-1,x)+2x*v(n-1,x),
%F A208748 where u(1,x)=1, v(1,x)=1.
%F A208748 T(n,k) = 2^k*A208343(n,k). - _Philippe Deléham_, Mar 05 2012
%F A208748 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) =   1, T(2,0) = T(3,0) = 0, T(2,1) = 4, T(3,1) = 2, T(3,2) = 12, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 14 2012
%F A208748 G.f.: (-1+x-2*x*y)*x*y/(-1+x+2*x*y-2*x^2*y+4*x^2*y^2). - _R. J. Mathar_, Aug 11 2015
%e A208748 First five rows:
%e A208748 1
%e A208748 0...4
%e A208748 0...2...12
%e A208748 0...2...8...40
%e A208748 0...2...8...40...128
%e A208748 First five polynomials v(n,x):
%e A208748 1
%e A208748 4x
%e A208748 2x + 12x^2
%e A208748 2x + 8x^2 + 40x^3
%e A208748 2x + 8x^2 + 40x^3 + 128x^4
%t A208748 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208748 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208748 v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x];
%t A208748 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208748 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208748 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208748 TableForm[cu]
%t A208748 Flatten[%]    (* A208747 *)
%t A208748 Table[Expand[v[n, x]], {n, 1, z}]
%t A208748 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208748 TableForm[cv]
%t A208748 Flatten[%]    (* A208748 *)
%Y A208748 Cf. A208747, A208510.
%K A208748 nonn,tabl
%O A208748 1,3
%A A208748 _Clark Kimberling_, Mar 02 2012