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A209687 Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section.

%I A209687 #13 Aug 12 2015 14:43:09
%S A209687 1,0,2,0,1,5,0,1,5,12,0,1,6,18,29,0,1,7,26,58,70,0,1,8,35,98,175,169,
%T A209687 0,1,9,45,149,339,507,408,0,1,10,56,212,574,1108,1428,985,0,1,11,68,
%U A209687 288,894,2066,3476,3940,2378,0,1,12,81,378,1314,3492,7074,10572
%N A209687 Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section.
%C A209687 Alternating row sums: signed powers of 2.  For a discussion and guide to related arrays, see A208510.
%C A209687 As triangle T(n,k) with 0<=k<=n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 28 2012
%F A209687 u(n,x)=x*u(n-1,x)+x*v(n-1,x),
%F A209687 v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x),
%F A209687 where u(1,x)=1, v(1,x)=1.
%F A209687 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) = 2 and T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 28 2012
%F A209687 G.f.: (-1+x)/(-1+x+2*x*y-x^2*y+x^2*y^2). - _R. J. Mathar_, Aug 12 2015
%e A209687 First five rows:
%e A209687 1
%e A209687 0...2
%e A209687 0...1...5
%e A209687 0...1...5...12
%e A209687 0...1...6...18...29
%e A209687 First three polynomials v(n,x): 1, 2x, x + 5x^2.
%t A209687 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209687 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
%t A209687 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209687 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209687 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209687 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209687 TableForm[cu]
%t A209687 Flatten[%]   (* A209687 *)
%t A209687 Table[Expand[v[n, x]], {n, 1, z}]
%t A209687 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209687 TableForm[cv]
%t A209687 Flatten[%]   (* A208339 *)
%Y A209687 Cf. A208339, A208510.
%K A209687 nonn,tabl
%O A209687 1,3
%A A209687 _Clark Kimberling_, Mar 12 2012