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

%I A209566 #6 Mar 30 2012 18:58:15
%S A209566 1,1,3,1,3,7,1,3,10,15,1,3,10,31,31,1,3,10,34,88,63,1,3,10,34,113,233,
%T A209566 127,1,3,10,34,116,359,586,255,1,3,10,34,116,393,1084,1419,511,1,3,10,
%U A209566 34,116,396,1306,3120,3340,1023,1,3,10,34,116,396,1349,4216
%N A209566 Triangle of coefficients of polynomials v(n,x) jointly generated with A209565; see the Formula section.
%C A209566  For a discussion and guide to related arrays, see A208510.
%F A209566 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209566 v(n,x)=x*u(n-1,x)+2x*v(n-1,x) +1,
%F A209566 where u(1,x)=1, v(1,x)=1.
%e A209566 First five rows:
%e A209566 1
%e A209566 1...1
%e A209566 1...4...1
%e A209566 1...4...11...1
%e A209566 1...4...14...26...1
%e A209566 First three polynomials v(n,x): 1, 1 + x , 1 + 4x + x^2 .
%t A209566 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209566 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209566 v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209566 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209566 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209566 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209566 TableForm[cu]
%t A209566 Flatten[%]   (* A209565 *)
%t A209566 Table[Expand[v[n, x]], {n, 1, z}]
%t A209566 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209566 TableForm[cv]
%t A209566 Flatten[%]   (* A209566 *)
%Y A209566 Cf. A209565, A208510.
%K A209566 nonn,tabl
%O A209566 1,3
%A A209566 _Clark Kimberling_, Mar 10 2012