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A210041 Triangle of coefficients of polynomials u(n,x) jointly generated with A209758; see the Formula section.

%I A210041 #10 Apr 20 2012 21:20:19
%S A210041 1,3,5,3,7,8,6,9,15,19,12,11,24,41,44,24,13,35,74,107,100,48,15,48,
%T A210041 120,214,270,224,96,17,63,181,380,591,664,496,192,19,80,259,622,1137,
%U A210041 1576,1600,1088,384,21,99,356,959,2001,3259,4088,3792,2368,768,23
%N A210041 Triangle of coefficients of polynomials u(n,x) jointly generated with A209758; see the Formula section.
%C A210041 Row sums: even-indexed Fibonacci numbers: 1,3,8,21,55,...
%C A210041 For a discussion and guide to related arrays, see A208510.
%F A210041 u(n,x)=u(n-1,x)+v(n-1,x)+1,
%F A210041 v(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
%F A210041 where u(1,x)=1, v(1,x)=1.
%e A210041 First five rows:
%e A210041 1
%e A210041 3
%e A210041 5...3
%e A210041 7...8....6
%e A210041 9...15...19...12
%e A210041 First three polynomials u(n,x): 1, 3, 5 + 3x.
%t A210041 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210041 u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A210041 v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A210041 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210041 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210041 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210041 TableForm[cu]
%t A210041 Flatten[%]    (* A210041 *)
%t A210041 Table[Expand[v[n, x]], {n, 1, z}]
%t A210041 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210041 TableForm[cv]
%t A210041 Flatten[%]    (* A209758 *)
%Y A210041 Cf. A209758, A208510.
%K A210041 nonn,tabf
%O A210041 1,2
%A A210041 _Clark Kimberling_, Mar 17 2012