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A209745 Triangle of coefficients of polynomials u(n,x) jointly generated with A209746; see the Formula section.

%I A209745 #13 Sep 08 2013 19:59:32
%S A209745 1,1,2,2,5,4,3,12,16,8,5,25,49,44,16,8,50,127,166,112,32,13,96,301,
%T A209745 513,504,272,64,21,180,670,1408,1808,1424,640,128,34,331,1427,3562,
%U A209745 5641,5816,3824,1472,256,55,600,2939,8494,15981,20330,17520,9888
%N A209745 Triangle of coefficients of polynomials u(n,x) jointly generated with A209746; see the Formula section.
%C A209745 Row n begins with F(n) and ends with 2^(n-1), where F=A000045 (Fibonacci numbers)
%C A209745 Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...
%C A209745 For a discussion and guide to related arrays, see A208510.
%C A209745 Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ....) DELTA (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 24 2012
%C A209745 Riordan array (1/(1-x-x^2), (2*x+x^2)/(1-x-x^2)). - _Philippe Deléham_, Mar 24 2012
%F A209745 u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
%F A209745 v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x),
%F A209745 where u(1,x)=1, v(1,x)=1.
%F A209745 T(n,k) = T(n-1, k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 24 2012
%e A209745 First five rows:
%e A209745 1
%e A209745 1...2
%e A209745 2...5....4
%e A209745 3...12...16...8
%e A209745 5...25...49...44...16
%e A209745 First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.
%e A209745 (0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 1, 0, 0, 0, ...) begins :
%e A209745 1
%e A209745 0, 1
%e A209745 0, 1, 2
%e A209745 0, 2, 5, 4
%e A209745 0, 3, 12, 16, 8
%e A209745 0, 5, 25, 49, 44, 16 ... - _Philippe Deléham_, Mar 24 2012
%t A209745 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209745 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209745 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209745 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209745 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209745 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209745 TableForm[cu]
%t A209745 Flatten[%]    (* A209745 *)
%t A209745 Table[Expand[v[n, x]], {n, 1, z}]
%t A209745 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209745 TableForm[cv]
%t A209745 Flatten[%]    (* A209746 *)
%Y A209745 Cf. A000045, A000079, A209746, A208510.
%K A209745 nonn,tabl
%O A209745 1,3
%A A209745 _Clark Kimberling_, Mar 13 2012