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A209137 Triangle of coefficients of polynomials u(n,x) jointly generated with A209138; see the Formula section. 4

%I

%S 1,2,1,3,4,2,5,10,9,3,8,22,28,18,5,13,45,74,68,35,8,21,88,177,210,154,

%T 66,13,34,167,397,574,541,331,122,21,55,310,850,1446,1656,1302,686,

%U 222,34,89,566,1758,3434,4614,4404,2982,1382,399,55,144,1020

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209138; see the Formula section.

%C Every row begins and ends with a Fibonacci number (A000045).

%C u(n,1) = n-th row sum = 3^(n-1).

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

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

%C Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 11 2012

%C Mirror image of triangle in A209138. - _Philippe Deléham_, Apr 11 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*v(n-1,x),

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

%F From _Philippe Deléham_, Apr 11 2012: (Start)

%F As DELTA-triangle T(n,k) with 0 <= k <= n:

%F G.f.: (1-y*x-y*x^2-y^2*x^2)/(1-x-y*x-x^2-y*x^2-y^2*x^2).

%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

%e First five rows:

%e 1;

%e 2, 1;

%e 3, 4, 2;

%e 5, 10, 9, 3;

%e 8, 22, 28, 18, 5;

%e First three polynomials u(n,x):

%e 1

%e 2 + x

%e 3 + 4x + 2x^2

%e From _Philippe Deléham_, Apr 11 2012: (Start)

%e (1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:

%e 1;

%e 1, 0;

%e 2, 1, 0;

%e 3, 4, 2, 0;

%e 5, 10, 9, 3, 0;

%e 8, 22, 28, 18, 5, 0;

%e 13, 45, 74, 68, 35, 8, 0; (End)

%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*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[%] (* A209137 *)

%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[%] (* A209138 *)

%Y Cf. A209138, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 05 2012

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Last modified June 18 11:58 EDT 2021. Contains 345098 sequences. (Running on oeis4.)