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A209153 Triangle of coefficients of polynomials u(n,x) jointly generated with A208340; see the Formula section.

%I A209153 #5 Mar 30 2012 18:58:15
%S A209153 1,2,1,4,5,2,7,14,11,3,11,31,38,22,5,16,60,103,93,43,8,22,106,239,298,
%T A209153 212,81,13,29,175,497,802,782,459,150,21,37,274,952,1909,2393,1917,
%U A209153 958,273,34,46,411,1710,4143,6410,6570,4465,1942,491,55,56,595
%N A209153 Triangle of coefficients of polynomials u(n,x) jointly generated with A208340; see the Formula section.
%C A209153 Every row ends with a Fibonacci number (A000045).
%C A209153 Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
%C A209153 For a discussion and guide to related arrays, see A208510.
%F A209153 u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F A209153 v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A209153 where u(1,x)=1, v(1,x)=1.
%e A209153 First five rows:
%e A209153 1
%e A209153 2....1
%e A209153 4....5....2
%e A209153 7....14...11...3
%e A209153 11...31...38...22...5
%e A209153 First three polynomials v(n,x): 1, 2 + x, 4 + 5x + 2x^2.
%t A209153 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209153 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209153 v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209153 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209153 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209153 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209153 TableForm[cu]
%t A209153 Flatten[%]    (* A209153 *)
%t A209153 Table[Expand[v[n, x]], {n, 1, z}]
%t A209153 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209153 TableForm[cv]
%t A209153 Flatten[%]    (* A208340 *)
%Y A209153 Cf. A208340, A208510.
%K A209153 nonn,tabl
%O A209153 1,2
%A A209153 _Clark Kimberling_, Mar 07 2012