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%I #12 Sep 08 2013 19:59:26
%S 1,1,2,1,6,4,1,10,16,8,1,14,36,40,16,1,18,64,112,96,32,1,22,100,240,
%T 320,224,64,1,26,144,440,800,864,512,128,1,30,196,728,1680,2464,2240,
%U 1152,256,1,34,256,1120,3136,5824,7168,5632,2560,512,1,38,324
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208659; see the Formula section.
%C Alternating row sums: 1,-1,-1,-1,-1,-1,-1,-1,-1,...
%C For a discussion and guide to related arrays, see A208510.
%F u(n,x)=u(n-1,x)+2x*v(n-1,x),
%F v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%F 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) = T(2,0) = T(3,0) = 1, T(2,1) = 2, T(3,1) = 6, T(3,2) = 4, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 19 2012
%e First five rows:
%e 1
%e 1...2
%e 1...6...4
%e 1...10...16...8
%e 1...14...36...40...16
%e First five polynomials u(n,x):
%e 1
%e 1 + 2x
%e 1 + 6x + 4x^2
%e 1 + 10x + 16x^2 + 8x^3
%e 1 + 14x + 36x^2 + 40x^3 + 16x^4
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t v[n_, x_] := u[n - 1, x] + 2 x*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[%] (* A185045 *)
%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[%] (* A208659 *)
%Y Cf. A208659, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 03 2012
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