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Triangle of coefficients of polynomials v(n,x) jointly generated with A210789; see the Formula section.
3

%I #17 Jan 27 2020 01:33:38

%S 1,1,2,1,2,3,1,4,5,5,1,4,10,10,8,1,6,14,24,20,13,1,6,21,38,52,38,21,1,

%T 8,27,65,96,109,71,34,1,8,36,92,176,224,220,130,55,1,10,44,136,280,

%U 446,500,434,235,89,1,10,55,180,440,772,1066,1074,839,420,144,1

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210789; see the Formula section.

%C Row n starts with 1 and ends with F(n+1), where F=A000045 (Fibonacci numbers).

%C Column 2: 2,2,4,4,6,6,8,8,...

%C Row sums: A105476.

%C Alternating row sums: signed Fibonacci numbers.

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

%C Subtriangle of the triangle given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 28 2012

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

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

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

%F From _Philippe Deléham_, Mar 28 2012: (Start)

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

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

%F T(n,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(2,0) = 1, T(2,1) = 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 1, 2;

%e 1, 2, 3;

%e 1, 4, 5, 5;

%e 1, 4, 10, 10, 8;

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

%e 1

%e 1 + 2x

%e 1 + 2x + 3x^2.

%e From _Philippe Deléham_, Mar 28 2012: (Start)

%e (1, 0, -1, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins:

%e 1;

%e 1, 0;

%e 1, 2, 0;

%e 1, 2, 3, 0;

%e 1, 4, 5, 5, 0;

%e 1, 4, 10, 10, 8, 0; (End)

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

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

%t d[x_] := h + x; e[x_] := p + x;

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

%t j = 0; c = 0; h = 2; p = -1; f = 0;

%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[%] (* A210789 *)

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

%t TableForm[cv]

%t Flatten[%] (* A210790 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A006138 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A105476 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *)

%Y Cf. A210789, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 26 2012