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

%I A208517 #6 Mar 30 2012 18:58:13
%S A208517 1,1,3,1,3,7,1,3,8,17,1,3,9,22,41,1,3,10,27,59,99,1,3,11,32,78,157,
%T A208517 239,1,3,12,37,98,222,414,577,1,3,13,42,119,294,622,1084,1393,1,3,14,
%U A208517 47,141,373,864,1722,2821,3363,1,3,15,52,164,459,1141,2500,4719
%N A208517 Triangle of coefficients of polynomials v(n,x) jointly generated with A208516; see the Formula section.
%F A208517 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208517 v(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
%F A208517 where u(1,x)=1, v(1,x)=1.
%e A208517 First five rows:
%e A208517 1
%e A208517 1...3
%e A208517 1...3...7
%e A208517 1...3...8...17
%e A208517 1...3...9...22...41
%e A208517 First five polynomials v(n,x):
%e A208517 1
%e A208517 1 + 3x
%e A208517 1 + 3x + 7x^2
%e A208517 1 + 3x + 8x^2 + 17x^3
%e A208517 1 + 3x + 9x^2 + 22x^3 + 41x^4
%t A208517 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208517 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208517 v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A208517 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208517 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208517 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208517 TableForm[cu]
%t A208517 Flatten[%]  (* A208516 *)
%t A208517 Table[Expand[v[n, x]], {n, 1, z}]
%t A208517 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208517 TableForm[cv]
%t A208517 Flatten[%]  (* A208517 *)
%Y A208517 Cf. A208514.
%K A208517 nonn,tabl
%O A208517 1,3
%A A208517 _Clark Kimberling_, Feb 28 2012