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

%I A209167 #10 Aug 07 2025 04:35:18
%S A209167 1,2,2,3,5,3,5,12,12,5,8,25,35,25,8,13,50,89,89,50,13,21,96,207,263,
%T A209167 207,96,21,34,180,455,698,698,455,180,34,55,331,959,1719,2073,1719,
%U A209167 959,331,55,89,600,1959,4011,5643,5643,4011,1959,600,89,144,1075
%N A209167 Triangle of coefficients of polynomials v(n,x) jointly generated with A209166; see the Formula section.
%C A209167 Row n begins and ends with F(n+1), where F=A000045 (Fibonacci numbers).
%C A209167 Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,1,0,...
%C A209167 For a discussion and guide to related arrays, see A208510.
%F A209167 u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F A209167 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A209167 where u(1,x)=1, v(1,x)=1.
%e A209167 First five rows:
%e A209167   1
%e A209167   2    2
%e A209167   3    5    3
%e A209167   5    12   12   5
%e A209167   8    25   35   25   8
%e A209167 First three polynomials v(n,x): 1, 2 + 2*x, 3 + 5*x + 3*x^2.
%t A209167 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209167 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209167 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209167 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209167 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209167 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209167 TableForm[cu]
%t A209167 Flatten[%]    (* A209166 *)
%t A209167 Table[Expand[v[n, x]], {n, 1, z}]
%t A209167 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209167 TableForm[cv]
%t A209167 Flatten[%]    (* A209167 *)
%Y A209167 Cf. A209166, A208510.
%K A209167 nonn,tabl
%O A209167 1,2
%A A209167 _Clark Kimberling_, Mar 08 2012