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A208516 Triangle of coefficients of polynomials u(n,x) jointly generated with A208517; see the Formula section.

%I A208516 #5 Mar 30 2012 18:58:13
%S A208516 1,1,1,1,2,3,1,3,6,7,1,4,9,15,17,1,5,12,24,39,41,1,6,15,34,66,100,99,
%T A208516 1,7,18,45,98,178,256,239,1,8,21,57,135,276,478,653,577,1,9,24,70,177,
%U A208516 395,772,1275,1661,1393,1,10,27,84,224,536,1145,2139,3383,4214
%N A208516 Triangle of coefficients of polynomials u(n,x) jointly generated with A208517; see the Formula section.
%F A208516 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208516 v(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
%F A208516 where u(1,x)=1, v(1,x)=1.
%e A208516 First five rows:
%e A208516 1
%e A208516 1...1
%e A208516 1...2...3
%e A208516 1...3...6...7
%e A208516 1...4...9...15...17
%e A208516 First five polynomials u(n,x):
%e A208516 1
%e A208516 1 + x
%e A208516 1 + 2x + 3x^2
%e A208516 1 + 3x + 6x^2 + 7x^3
%e A208516 1 + 4x + 9x^2 + 15x^3 + 17x^4
%t A208516 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208516 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208516 v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A208516 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208516 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208516 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208516 TableForm[cu]
%t A208516 Flatten[%]  (* A208516 *)
%t A208516 Table[Expand[v[n, x]], {n, 1, z}]
%t A208516 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208516 TableForm[cv]
%t A208516 Flatten[%]  (* A208517 *)
%Y A208516 Cf. A208517.
%K A208516 nonn,tabl
%O A208516 1,5
%A A208516 _Clark Kimberling_, Feb 28 2012