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A209166 Triangle of coefficients of polynomials u(n,x) jointly generated with A209167; see the Formula section.

%I A209166 #6 Mar 30 2012 18:58:15
%S A209166 1,2,1,4,5,2,7,13,10,3,12,30,34,20,5,20,63,94,80,38,8,33,126,233,258,
%T A209166 177,71,13,54,243,536,728,647,374,130,21,88,457,1171,1881,2043,1527,
%U A209166 765,235,34,143,843,2461,4559,5835,5319,3443,1525,420,55,232
%N A209166 Triangle of coefficients of polynomials u(n,x) jointly generated with A209167; see the Formula section.
%C A209166 Row n begins with F(n+2)-1 and ends with F(n), where F=A000045 (Fibonacci numbers).
%C A209166 Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
%C A209166 For a discussion and guide to related arrays, see A208510.
%F A209166 u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F A209166 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A209166 where u(1,x)=1, v(1,x)=1.
%e A209166 First five rows:
%e A209166 1
%e A209166 2....1
%e A209166 5....5....1
%e A209166 11...15...6....1
%e A209166 23...41...27...9...1
%e A209166 First three polynomials v(n,x): 1, 2 + x, 5 + 5x + x^2.
%t A209166 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209166 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209166 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209166 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209166 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209166 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209166 TableForm[cu]
%t A209166 Flatten[%]    (* A209166 *)
%t A209166 Table[Expand[v[n, x]], {n, 1, z}]
%t A209166 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209166 TableForm[cv]
%t A209166 Flatten[%]    (* A209167 *)
%Y A209166 Cf. A209167, A208510.
%K A209166 nonn,tabl
%O A209166 1,2
%A A209166 _Clark Kimberling_, Mar 08 2012