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A209831 Triangle of coefficients of polynomials v(n,x) jointly generated with A209830; see the Formula section.

%I A209831 #14 Jan 26 2020 21:06:02
%S A209831 1,1,3,1,5,8,1,8,20,21,1,10,41,71,55,1,13,65,176,235,144,1,15,99,338,
%T A209831 684,744,377,1,18,135,590,1536,2490,2285,987,1,20,182,926,3031,6382,
%U A209831 8651,6865,2584,1,23,230,1388,5359,14065,24875,29020,20284,6765
%N A209831 Triangle of coefficients of polynomials v(n,x) jointly generated with A209830; see the Formula section.
%C A209831 Each row begins with 1 and ends with an even-indexed Fibonacci number.
%C A209831 Alternating row sums: signed powers of 2.
%C A209831 For a discussion and guide to related arrays, see A208510.
%C A209831 Subtriangle of the triangle given by (1, 0, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 16 2012
%F A209831 u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
%F A209831 v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
%F A209831 where u(1,x)=1, v(1,x)=1.
%F A209831 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209831 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) = 3 and T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Mar 16 2012
%F A209831 As DELTA-triangle with 0 <= k <= n: g.f.: (1 + x - 3*y*x - 2*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
%e A209831 From _Philippe Deléham_, Mar 16 2012: (Start)
%e A209831 First five rows:
%e A209831   1;
%e A209831   1,  3;
%e A209831   1,  5,  8;
%e A209831   1,  8, 20, 21;
%e A209831   1, 10, 41, 71, 55;
%e A209831 First three polynomials v(n,x):
%e A209831   1
%e A209831   1 + 3x
%e A209831   1 + 5x + 8x^2.
%e A209831 (1, 0, -1/3, -2/3, 0, 0, ...) DELTA (0, 3, -1/3, 1/3, 0, 0, ...) begins:
%e A209831   1;
%e A209831   1,  0;
%e A209831   1,  3,  0;
%e A209831   1,  5,  8,  0;
%e A209831   1,  8, 20, 21,  0;
%e A209831   1, 10, 41, 71, 55,  0; (End)
%t A209831 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209831 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209831 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
%t A209831 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209831 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209831 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209831 TableForm[cu]
%t A209831 Flatten[%]    (* A209830 *)
%t A209831 Table[Expand[v[n, x]], {n, 1, z}]
%t A209831 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209831 TableForm[cv]
%t A209831 Flatten[%]    (* A209831 *)
%Y A209831 Cf. A209830, A208510.
%K A209831 nonn,tabl
%O A209831 1,3
%A A209831 _Clark Kimberling_, Mar 13 2012