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A210563 Triangle of coefficients of polynomials u(n,x) jointly generated with A210564; see the Formula section.

%I A210563 #5 Mar 30 2012 18:58:17
%S A210563 1,1,2,1,3,5,1,3,10,13,1,3,11,32,34,1,3,11,40,99,89,1,3,11,41,141,299,
%T A210563 233,1,3,11,41,152,482,887,610,1,3,11,41,153,556,1604,2595,1597,1,3,
%U A210563 11,41,153,570,1998,5217,7508,4181,1,3,11,41,153,571,2113,7042
%N A210563 Triangle of coefficients of polynomials u(n,x) jointly generated with A210564; see the Formula section.
%C A210563 Last terms in rows: odd-indexed Fibonacci numbers
%C A210563 Limiting row:  A001835
%C A210563 For a discussion and guide to related arrays, see A208510.
%F A210563 u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F A210563 v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
%F A210563 where u(1,x)=1, v(1,x)=1.
%e A210563 First five rows:
%e A210563 1
%e A210563 1...2
%e A210563 1...3...5
%e A210563 1...3...10...13
%e A210563 1...3...11...32...34
%e A210563 First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 5x^2.
%t A210563 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210563 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210563 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A210563 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210563 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210563 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210563 TableForm[cu]
%t A210563 Flatten[%]  (* A210563 *)
%t A210563 Table[Expand[v[n, x]], {n, 1, z}]
%t A210563 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210563 TableForm[cv]
%t A210563 Flatten[%]  (* A210564 *)
%Y A210563 Cf. A210564, A208510.
%K A210563 nonn,tabl
%O A210563 1,3
%A A210563 _Clark Kimberling_, Mar 23 2012