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%I #21 Jan 27 2020 01:34:20
%S 1,1,1,1,2,2,1,3,4,3,1,4,8,8,5,1,5,12,18,15,8,1,6,18,32,39,28,13,1,7,
%T 24,53,77,80,51,21,1,8,32,80,142,176,160,92,34,1,9,40,116,234,352,384,
%U 312,164,55,1,10,50,160,370,632,830,812,598,290,89,1,11,60,215
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210790; see the Formula section.
%C Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).
%C Column 2: 1,2,3,4,5,6,7,8,...
%C Row sums: A006138.
%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, 0, -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_, 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*x^2-y^2*x^2)/(1-y*x-y*x^2-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) = T(2,1) = 1, 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, 1;
%e 1, 2, 2;
%e 1, 3, 4, 3;
%e 1, 4, 8, 8, 5;
%e First three polynomials u(n,x):
%e 1
%e 1 + x
%e 1 + 2x + 2x^2.
%e From _Philippe Deléham_, Mar 28 2012: (Start)
%e (1, 0, 0, -1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 2, 2, 0;
%e 1, 3, 4, 3, 0;
%e 1, 4, 8, 8, 5, 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. A210790, A208510.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Mar 26 2012
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