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A209705 Triangle of coefficients of polynomials u(n,x) jointly generated with A209706; see the Formula section.

%I A209705 #9 Dec 28 2013 04:14:15
%S A209705 1,0,2,0,3,4,0,4,10,8,0,5,18,28,16,0,6,28,64,72,32,0,7,40,120,200,176,
%T A209705 64,0,8,54,200,440,576,416,128,0,9,70,308,840,1456,1568,960,256,0,10,
%U A209705 88,448,1456,3136,4480,4096,2176,512,0,11,108,624,2352,6048
%N A209705 Triangle of coefficients of polynomials u(n,x) jointly generated with A209706; see the Formula section.
%C A209705 Alternating row sums: 1,-2,1,-2,1,-2,1,-2,...
%C A209705 For a discussion and guide to related arrays, see A208510.
%F A209705 u(n,x) = x*u(n-1,x)+x*v(n-1,x),
%F A209705 v(n,x) = (x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A209705 where u(1,x)=1, v(1,x)=1.
%F A209705 T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)-T(n-2,k)-2*T(n-2,k-1), T(1,0)=1, T(2,0)=0, T(2,1)=2, T(3,0)=0, T(3,1)=3, T(3,2)=4, T(n,k)=0 if k<0 or if k>=n. - _Philippe Deléham_, Dec 27 2013
%e A209705 First five rows:
%e A209705 1
%e A209705 0...2
%e A209705 0...3...4
%e A209705 0...4...10...8
%e A209705 0...5...18...28...16
%e A209705 First three polynomials v(n,x): 1, 2x, 3x + 4x^2.
%t A209705 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209705 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
%t A209705 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209705 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209705 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209705 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209705 TableForm[cu]
%t A209705 Flatten[%]  (* A209705 *)
%t A209705 Table[Expand[v[n, x]], {n, 1, z}]
%t A209705 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209705 TableForm[cv]
%t A209705 Flatten[%]  (* A209706 *)
%Y A209705 Cf. A209706, A208510.
%K A209705 nonn,tabl
%O A209705 1,3
%A A209705 _Clark Kimberling_, Mar 12 2012