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Triangle of coefficients of polynomials u(n,x) jointly generated with A209706; see the Formula section.
4

%I #9 Dec 28 2013 04:14:15

%S 1,0,2,0,3,4,0,4,10,8,0,5,18,28,16,0,6,28,64,72,32,0,7,40,120,200,176,

%T 64,0,8,54,200,440,576,416,128,0,9,70,308,840,1456,1568,960,256,0,10,

%U 88,448,1456,3136,4480,4096,2176,512,0,11,108,624,2352,6048

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209706; see the Formula section.

%C Alternating row sums: 1,-2,1,-2,1,-2,1,-2,...

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

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

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

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

%F T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)-T(n-2,k)-2*T(n-2,k-1), T(1,0)=1, T(2,0)=0, T(2,1)=2, T(3,0)=0, T(3,1)=3, T(3,2)=4, T(n,k)=0 if k<0 or if k>=n. - _Philippe Deléham_, Dec 27 2013

%e First five rows:

%e 1

%e 0...2

%e 0...3...4

%e 0...4...10...8

%e 0...5...18...28...16

%e First three polynomials v(n,x): 1, 2x, 3x + 4x^2.

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

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

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

%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[%] (* A209705 *)

%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[%] (* A209706 *)

%Y Cf. A209706, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 12 2012