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A210871 Triangle of coefficients of polynomials v(n,x) jointly generated with A210870; see the Formula section.

%I A210871 #5 Mar 31 2012 20:30:51
%S A210871 1,1,2,1,1,3,1,3,2,5,1,2,7,3,8,1,4,5,15,5,13,1,3,12,10,30,8,21,1,5,9,
%T A210871 31,20,58,13,34,1,4,18,22,73,38,109,21,55,1,6,14,54,51,162,71,201,34,
%U A210871 89,1,5,25,40,145,111,344,130,365,55,144,1,7,20,85,105,361,233
%N A210871 Triangle of coefficients of polynomials v(n,x) jointly generated with A210870; see the Formula section.
%C A210871 Row n, for n>2, starts with 1 and A028242(n) and ends with F(n-1) and F(n+1), where F=A000045 (Fibonacci numbers).
%C A210871 Row sums:  A001045
%C A210871 Alternating row sums: A077925
%C A210871 For a discussion and guide to related arrays, see A208510.
%F A210871 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A210871 v(n,x)=(x+1)*u(n-1,x)+(x-1)*v(n-1,x)+1,
%F A210871 where u(1,x)=1, v(1,x)=1.
%e A210871 First six rows:
%e A210871 1
%e A210871 1...2
%e A210871 1...1...3
%e A210871 1...3...2....5
%e A210871 1...2...7....3....8
%e A210871 1...4...5....15...5...13
%e A210871 First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2
%t A210871 u[1, x_] := 1; v[1, x_] := 1; z = 14;
%t A210871 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A210871 v[n_, x_] := (x + 1)*u[n - 1, x] + (x - 1)*v[n - 1, x] + 1;
%t A210871 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210871 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210871 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210871 TableForm[cu]
%t A210871 Flatten[%]    (* A210870 *)
%t A210871 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210871 TableForm[cv]
%t A210871 Flatten[%]    (* A210871 *)
%t A210871 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000975 *)
%t A210871 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A001045 *)
%t A210871 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A113954 *)
%t A210871 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
%Y A210871 Cf. A210870, A208510.
%K A210871 nonn,tabl
%O A210871 1,3
%A A210871 _Clark Kimberling_, Mar 29 2012