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A210876 Triangle of coefficients of polynomials u(n,x) jointly generated with A210877; see the Formula section.

%I A210876 #5 Oct 02 2013 16:26:12
%S A210876 1,2,1,1,5,1,1,4,9,1,1,3,12,14,1,1,3,9,29,20,1,1,3,8,27,60,27,1,1,3,8,
%T A210876 22,74,111,35,1,1,3,8,21,63,181,189,44,1,1,3,8,21,56,178,399,302,54,1,
%U A210876 1,3,8,21,55,154,474,806,459,65,1,1,3,8,21,55,145,430,1169
%N A210876 Triangle of coefficients of polynomials u(n,x) jointly generated with A210877; see the Formula section.
%C A210876 For n>2, each row begins with 1 and ends with 1.  If the term in row n and column k is denoted by U(n,k), then U(n,n-2)=A000096(n-1) and U(n,n-3)=A086274(n-1).
%C A210876 Row sums: A000225 (-1+2^n)
%C A210876 Alternating row sums:  A077973
%C A210876 Limiting row:  1,3,8,21,55,..., even-indexed Fibonacci numbers
%C A210876 For a discussion and guide to related arrays, see A208510.
%F A210876 u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F A210876 v(n,x)=x*u(n-1,x)+x*v(n-1,x)+x,
%F A210876 where u(1,x)=1, v(1,x)=1.
%e A210876 First six rows:
%e A210876 1
%e A210876 2...1
%e A210876 1...5...1
%e A210876 1...4...9....1
%e A210876 1...3...12...14...1
%e A210876 1...3...9....29...20...1
%e A210876 First three polynomials u(n,x): 1, 2 + x, 1 + 5x + x^2.
%t A210876 u[1, x_] := 1; v[1, x_] := 1; z = 14;
%t A210876 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A210876 v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + x;
%t A210876 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210876 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210876 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210876 TableForm[cu]
%t A210876 Flatten[%]    (* A210876 *)
%t A210876 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210876 TableForm[cv]
%t A210876 Flatten[%]    (* A210877 *)
%t A210876 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210876 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
%t A210876 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077973 *)
%t A210876 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
%Y A210876 Cf. A210877, A208510.
%K A210876 nonn,tabl
%O A210876 1,2
%A A210876 _Clark Kimberling_, Mar 30 2012