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

%I A210870 #5 Mar 31 2012 20:30:51
%S A210870 1,1,1,1,2,2,1,3,3,3,1,4,6,5,5,1,5,8,12,8,8,1,6,12,17,23,13,13,1,7,15,
%T A210870 29,33,43,21,21,1,8,20,38,64,63,79,34,34,1,9,24,56,86,136,117,143,55,
%U A210870 55,1,10,30,70,140,187,279,214,256,89,89,1,11,35,95,180,332
%N A210870 Triangle of coefficients of polynomials u(n,x) jointly generated with A210871; see the Formula section.
%C A210870 In row n, for n>1, the first two terms are 1 and n-1, and the last two are F(n) and F(n), where F = A000045 (Fibonacci numbers).
%C A210870 Row sums: A000975
%C A210870 Alternating row sums:  A113954
%C A210870 For a discussion and guide to related arrays, see A208510.
%F A210870 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A210870 v(n,x)=(x+1)*u(n-1,x)+(x-1)*v(n-1,x)+1,
%F A210870 where u(1,x)=1, v(1,x)=1.
%e A210870 First six rows:
%e A210870 1
%e A210870 1...1
%e A210870 1...2...2
%e A210870 1...3...3...3
%e A210870 1...4...6...5....5
%e A210870 1...5...8...12...8...8
%e A210870 First three polynomials u(n,x): 1, 1 + x, 1 + 2x + 2x^2.
%t A210870 u[1, x_] := 1; v[1, x_] := 1; z = 14;
%t A210870 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A210870 v[n_, x_] := (x + 1)*u[n - 1, x] + (x - 1)*v[n - 1, x] + 1;
%t A210870 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210870 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210870 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210870 TableForm[cu]
%t A210870 Flatten[%]    (* A210870 *)
%t A210870 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210870 TableForm[cv]
%t A210870 Flatten[%]    (* A210871 *)
%t A210870 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000975 *)
%t A210870 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A001045 *)
%t A210870 Table[u[n, x] /. x -> -1, {n, 1, z}]   (* A113954 *)
%t A210870 Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A077925 *)
%Y A210870 Cf. A210871, A208510.
%K A210870 nonn,tabl
%O A210870 1,5
%A A210870 _Clark Kimberling_, Mar 29 2012