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A209565 Triangle of coefficients of polynomials u(n,x) jointly generated with A209566; see the Formula section.

%I A209565 #7 Mar 30 2012 18:58:15
%S A209565 1,1,1,1,4,1,1,4,11,1,1,4,14,26,1,1,4,14,45,57,1,1,4,14,48,133,120,1,
%T A209565 1,4,14,48,161,366,247,1,1,4,14,48,164,520,952,502,1,1,4,14,48,164,
%U A209565 557,1604,2371,1013,1,1,4,14,48,164,560,1863,4724,5711,2036,1,1,4
%N A209565 Triangle of coefficients of polynomials u(n,x) jointly generated with A209566; see the Formula section.
%C A209565 For a discussion and guide to related arrays, see A208510.
%F A209565 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209565 v(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
%F A209565 where u(1,x)=1, v(1,x)=1.
%e A209565 First five rows:
%e A209565 1
%e A209565 1...1
%e A209565 1...4....1
%e A209565 1...4....11...1
%e A209565 1...4...14...26...1
%e A209565 First three polynomials v(n,x): 1, 1 + x, 1 + 4x + x^2.
%t A209565 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209565 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209565 v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209565 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209565 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209565 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209565 TableForm[cu]
%t A209565 Flatten[%]   (* A209565 *)
%t A209565 Table[Expand[v[n, x]], {n, 1, z}]
%t A209565 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209565 TableForm[cv]
%t A209565 Flatten[%]   (* A209566 *)
%Y A209565 Cf. A209566, A208510.
%K A209565 nonn,tabl
%O A209565 1,5
%A A209565 _Clark Kimberling_, Mar 10 2012