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%I #12 Aug 12 2015 04:05:10
%S 1,0,3,0,2,7,0,2,8,17,0,2,10,28,41,0,2,12,42,88,99,0,2,14,58,154,262,
%T 239,0,2,16,76,240,524,752,577,0,2,18,96,348,908,1692,2104,1393,0,2,
%U 20,118,480,1440,3224,5260,5776,3363,0,2,22,142,638,2148,5544
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209128; see the Formula section.
%C For a discussion and guide to related arrays, see A208510.
%C As triangle T(n,k) with 0<=k<=n, it is (0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 21 2012
%F u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F v(n,x)=x*u(n-1,x)+2x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 21 2012
%F G.f.: (-1+x-x*y)*x*y/(-1+x+2*x*y-x^2*y+x^2*y^2). - _R. J. Mathar_, Aug 12 2015
%e First five rows:
%e 1
%e 0...3
%e 0...2....7
%e 0...2....8...17
%e 0...2...10...28...41
%e First three polynomials v(n,x): 1, 3x, 2x + 7x^2.
%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*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[%] (* A209128 *)
%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[%] (* A209129 *)
%Y Cf. A209128, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 05 2012
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