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%I #19 Jan 24 2020 03:33:34
%S 1,3,2,6,8,3,12,25,19,5,24,68,77,40,8,48,172,259,201,80,13,96,416,782,
%T 806,478,154,21,192,976,2200,2825,2222,1067,289,34,384,2240,5888,9048,
%U 8857,5640,2277,532,55,768,5056,15184,27160,31787,25184,13483
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209170; see the Formula section.
%C Column 1: Fibonacci numbers (A000045).
%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 (1, 2, -3/2, 1/2, 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 10 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+1)*v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 3, T(2,1) = 2. - _Philippe Deléham_, Mar 10 2012
%F Sum_{k=0..n} T(n,k)*x^k = A000012(n), A003945(n-1), A007483(n-1) for x = -1, 0, 1 respectively. - _Philippe Deléham_, Mar 10 2012
%F G.f.: (-1-x-x*y)*x*y/(-1+2*x+x*y+x^2*y^2+x^2*y). - _R. J. Mathar_, Aug 12 2015
%e First five rows:
%e 1;
%e 3, 2;
%e 6, 8, 3;
%e 12, 25, 19, 5;
%e 24, 68, 77, 40, 8;
%e First three polynomials v(n,x):
%e 1
%e 3 + 2x
%e 6 + 8x + 3x^2.
%e From _Philippe Deléham_, Mar 10 2012: (Start)
%e Triangle (1, 2, -3/2, 1/2, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins (0 <= k <= n):
%e 1;
%e 1, 0;
%e 3, 2, 0;
%e 6, 8, 3, 0;
%e 12, 25, 19, 5, 0;
%e 24, 68, 77, 40, 8, 0;
%e 48, 172, 259, 201, 80, 13, 0;
%e 96, 416, 782, 806, 478, 154, 21, 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 + 1)*v[n - 1, x] + 1;
%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[%] (* A209170 *)
%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[%] (* A209171 *)
%Y Cf. A209170, A208510.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 08 2012
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