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A209142 Triangle of coefficients of polynomials v(n,x) jointly generated with A209141; see the Formula section.

%I A209142 #13 Aug 12 2015 04:09:30
%S A209142 1,2,2,4,7,3,8,20,17,5,16,52,65,37,8,32,128,210,176,75,13,64,304,616,
%T A209142 679,428,146,21,128,704,1696,2312,1921,971,276,34,256,1600,4464,7240,
%U A209142 7449,4970,2097,511,55,512,3584,11360,21344,26146,21622,12056
%N A209142 Triangle of coefficients of polynomials v(n,x) jointly generated with A209141; see the Formula section.
%C A209142 Each row begins with a power of 2 and ends with a Fibonacci number.
%C A209142 Alternating row sums: 1,0,0,0,0,0,0,0,0,0,0,...
%C A209142 For a discussion and guide to related arrays, see A208510.
%C A209142 As triangle T(n,k) with 0<=k<=n, it is (2, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 07 2012
%F A209142 u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F A209142 v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x),
%F A209142 where u(1,x)=1, v(1,x)=1.
%F A209142 As triangle T(n,k), 0<=k<=n : T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 07 2012
%F A209142 G.f.: (-1-x*y)*x*y/(-1+x*y+x^2*y^2+2*x+x^2*y). - _R. J. Mathar_, Aug 12 2015
%e A209142 First five rows:
%e A209142 1
%e A209142 2....2
%e A209142 4....7....3
%e A209142 8....20...17...5
%e A209142 16...52...65...37...8
%e A209142 First three polynomials v(n,x): 1, 2 + 2x, 4 + 7x + 3x^2.
%t A209142 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209142 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209142 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209142 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209142 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209142 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209142 TableForm[cu]
%t A209142 Flatten[%]    (* A209141 *)
%t A209142 Table[Expand[v[n, x]], {n, 1, z}]
%t A209142 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209142 TableForm[cv]
%t A209142 Flatten[%]    (* A209142 *)
%Y A209142 Cf. A209141, A208510.
%K A209142 nonn,tabl
%O A209142 1,2
%A A209142 _Clark Kimberling_, Mar 06 2012