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A210603 Triangle of coefficients of polynomials u(n,x) jointly generated with A210738; see the Formula section.

%I A210603 #5 Mar 30 2012 18:58:17
%S A210603 1,2,2,4,6,4,7,17,16,8,12,39,57,40,16,20,84,159,169,96,32,33,170,405,
%T A210603 551,465,224,64,54,332,950,1608,1727,1217,512,128,88,630,2115,4264,
%U A210603 5655,5055,3073,1152,256,143,1171,4515,10603,16666,18294,14079
%N A210603 Triangle of coefficients of polynomials u(n,x) jointly generated with A210738; see the Formula section.
%C A210603 Row n starts with F(n+2)-1, where F=A000045 (Fibonacci
%C A210603 numbers), and ends with 2^(n-1). For a discussion and
%C A210603 guide to related arrays, see A208510.
%F A210603 u(n,x)=2x*u(n-1,x)+v(n-1,x)+1,
%F A210603 v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A210603 where u(1,x)=1, v(1,x)=1.
%e A210603 First five rows:
%e A210603 1
%e A210603 2....2
%e A210603 4....6....4
%e A210603 7....17...16...8
%e A210603 12...39...57...40...16
%e A210603 First three polynomials u(n,x): 1, 2+ 2x, 4 + 6x + 4x^2.
%t A210603 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210603 u[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x] + 1;
%t A210603 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A210603 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210603 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210603 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210603 TableForm[cu]
%t A210603 Flatten[%]    (* A210603 *)
%t A210603 Table[Expand[v[n, x]], {n, 1, z}]
%t A210603 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210603 TableForm[cv]
%t A210603 Flatten[%]    (* A210738 *)
%Y A210603 Cf. A210738, A208510.
%K A210603 nonn,tabl
%O A210603 1,2
%A A210603 _Clark Kimberling_, Mar 24 2012