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Triangle of coefficients of polynomials v(n,x) jointly generated with A209141; see the Formula section.
3

%I #13 Aug 12 2015 04:09:30

%S 1,2,2,4,7,3,8,20,17,5,16,52,65,37,8,32,128,210,176,75,13,64,304,616,

%T 679,428,146,21,128,704,1696,2312,1921,971,276,34,256,1600,4464,7240,

%U 7449,4970,2097,511,55,512,3584,11360,21344,26146,21622,12056

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209141; see the Formula section.

%C Each row begins with a power of 2 and ends with a Fibonacci number.

%C Alternating row sums: 1,0,0,0,0,0,0,0,0,0,0,...

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

%C As triangle T(n,k) with 0<=k<=n, it is (2, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 07 2012

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

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

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

%F As triangle T(n,k), 0<=k<=n : T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 07 2012

%F G.f.: (-1-x*y)*x*y/(-1+x*y+x^2*y^2+2*x+x^2*y). - _R. J. Mathar_, Aug 12 2015

%e First five rows:

%e 1

%e 2....2

%e 4....7....3

%e 8....20...17...5

%e 16...52...65...37...8

%e First three polynomials v(n,x): 1, 2 + 2x, 4 + 7x + 3x^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_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%] (* A209141 *)

%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[%] (* A209142 *)

%Y Cf. A209141, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 06 2012