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

%I A209141 #13 Aug 12 2015 04:08:16
%S A209141 1,2,1,4,5,2,8,16,12,3,16,44,49,25,5,32,112,166,127,50,8,64,272,504,
%T A209141 513,301,96,13,128,640,1424,1808,1408,670,180,21,256,1472,3824,5816,
%U A209141 5641,3562,1427,331,34,512,3328,9888,17520,20330,15981,8494,2939
%N A209141 Triangle of coefficients of polynomials u(n,x) jointly generated with A209142; see the Formula section.
%C A209141 Each row begins with a power of 2 and ends with a Fibonacci number. Alternating row sums: all 1's.  For a discussion and guide to related arrays, see A208510.
%C A209141 Subtriangle of the triangle T(n,k) given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 07 2012
%F A209141 u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F A209141 v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x),
%F A209141 where u(1,x)=1, v(1,x)=1.
%F A209141 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) = 1, T(1,1) = 0,  T(n,k) = 0 if k>n or if k<0. - _Philippe Deléham_, Mar 07 2012
%F A209141 G.f.: -x*y/(-1+x*y+x^2*y^2+2*x+x^2*y). - _R. J. Mathar_, Aug 12 2015
%e A209141 First five rows:
%e A209141 1
%e A209141 2....1
%e A209141 4....5....2
%e A209141 8....16...12...3
%e A209141 16...44...49...25...5
%e A209141 First three polynomials u(n,x):  1, 2 + x, 4 + 5x + 2x^2
%e A209141 Triangle (1, 1, 0, 0, 0...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :
%e A209141 1
%e A209141 1, 0
%e A209141 2, 1, 0
%e A209141 4, 5, 2, 0
%e A209141 8, 16, 12, 3, 0
%e A209141 16, 44, 49, 25, 5, 0
%e A209141 32, 112, 166, 127, 50, 8, 0
%t A209141 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209141 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209141 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209141 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209141 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209141 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209141 TableForm[cu]
%t A209141 Flatten[%]    (* A209141 *)
%t A209141 Table[Expand[v[n, x]], {n, 1, z}]
%t A209141 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209141 TableForm[cv]
%t A209141 Flatten[%]    (* A209142 *)
%Y A209141 Cf. A209142, A208510.
%K A209141 nonn,tabl
%O A209141 1,2
%A A209141 _Clark Kimberling_, Mar 06 2012