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%I #19 Jan 24 2020 03:30:13
%S 1,1,2,1,5,5,1,7,18,13,1,10,35,59,34,1,12,61,147,185,89,1,15,90,302,
%T 558,564,233,1,17,129,527,1324,1986,1685,610,1,20,170,854,2653,5350,
%U 6761,4957,1597,1,22,222,1278,4811,12066,20383,22277,14406,4181,1
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209831; see the Formula section.
%C Each row begins with 1 and ends with an odd-indexed Fibonacci number.
%C Alternating row sums: 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, 0, 1/2, -3/2, 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 16 2012
%F u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
%F v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F As DELTA-triangle with 0 <= k <= n: G.f.: (1+x-3*y*x-3*y*x^2+y^2*x^2)/(1-3*y*x-x^2-2*y*x^2+y^2*x^2). - _Philippe Deléham_, Mar 16 2012
%F As DELTA-triangle: T(n,k) = 3*T(n-1,k-1) + T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1 = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Mar 16 2012
%e First five rows:
%e 1;
%e 1, 2;
%e 1, 5, 5;
%e 1, 7, 18, 13;
%e 1, 10, 35, 59, 34;
%e First three polynomials u(n,x):
%e 1
%e 1 + 2x
%e 1 + 5x + 5x^2.
%e From _Philippe Deléham_, Mar 16 2012: (Start)
%e (1, 0, 1/2, -3/2, 0, 0, ...) DELTA (0, 2, 1/2, 1/2, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 1, 2, 0;
%e 1, 5, 5, 0;
%e 1, 7, 18, 13, 0;
%e 1, 10, 35, 59, 34, 0; (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t v[n_, x_] := (x + 1)*u[n - 1, x] + 2 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[%] (* A209830 *)
%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[%] (* A209831 *)
%Y Cf. A209831, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 13 2012
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