A209134 - OEIS

%I #9 Sep 08 2013 19:59:31

%S 1,0,4,0,4,10,0,4,18,28,0,4,26,72,76,0,4,34,132,256,208,0,4,42,208,

%T 572,864,568,0,4,50,300,1056,2272,2808,1552,0,4,58,408,1740,4800,8496,

%U 8896,4240,0,4,66,532,2656,8880,20208,30432,27648,11584,0,4,74

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209133; see the Formula section.

%C For a discussion and guide to related arrays, see A208510.

%C As triangle T(n,k) with 0<=k<=n, it is (0, 1, 0, 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_, Apr 10 2012

%F u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),

%F v(n,x)=2x*u(n-1,x)+2x*v(n-1,x),

%F where u(1,x)=1, v(1,x)=1.

%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 4 and T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Apr 10 2012

%e First five rows:

%e 1

%e 0...4

%e 0...4...10

%e 0...4...18...28

%e 0...4...26...72...76

%e First three polynomials v(n,x): 1, 4x, 4x + 10x^2.

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];

%t v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x];

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]    (* A209133 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]    (* A209134 *)

%Y Cf. A209133, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 05 2012