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A210193 Triangle of coefficients of polynomials u(n,x) jointly generated with A210194; see the Formula section.

%I A210193 #4 Mar 30 2012 18:58:16
%S A210193 1,3,6,3,10,14,3,15,40,23,3,21,90,97,32,3,28,175,301,181,41,3,36,308,
%T A210193 770,728,292,50,3,45,504,1722,2346,1452,430,59,3,55,780,3486,6456,
%U A210193 5686,2554,595,68,3,66,1155,6534,15774,18722,11816,4115,787,77,3
%N A210193 Triangle of coefficients of polynomials u(n,x) jointly generated with A210194; see the Formula section.
%C A210193 Row sums:  powers of 3
%C A210193 For a discussion and guide to related arrays, see A208510.
%F A210193 u(n,x)=u(n-1,x)+v(n-1,x)+1,
%F A210193 v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A210193 where u(1,x)=1, v(1,x)=1.
%e A210193 First five rows:
%e A210193 1
%e A210193 3
%e A210193 6....3
%e A210193 10...14...3
%e A210193 15...40...23...3
%e A210193 First three polynomials u(n,x): 1, 3, 6 + 3x.
%t A210193 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210193 u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A210193 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A210193 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210193 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210193 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210193 TableForm[cu]
%t A210193 Flatten[%]  (* A210193 *)
%t A210193 Table[Expand[v[n, x]], {n, 1, z}]
%t A210193 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210193 TableForm[cv]
%t A210193 Flatten[%]  (* A210194 *)
%Y A210193 Cf. A210194, A208510.
%K A210193 nonn,tabf
%O A210193 1,2
%A A210193 _Clark Kimberling_, Mar 18 2012