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A210234 Triangle of coefficients of polynomials v(n,x) jointly generated with A210233; see the Formula section.

%I A210234 #4 Mar 30 2012 18:58:16
%S A210234 1,2,3,3,7,7,4,14,20,15,5,22,50,53,31,6,33,92,157,134,63,7,45,161,335,
%T A210234 455,327,127,8,60,248,666,1112,1248,776,255,9,76,372,1150,2466,3448,
%U A210234 3288,1801,511,10,95,520,1910,4732,8426,10144,8399,4106,1023,11
%N A210234 Triangle of coefficients of polynomials v(n,x) jointly generated with A210233; see the Formula section.
%C A210234 First and last terms of row n:  n and -1+2^n
%C A210234 Alternating row sums: 3^(n-1)
%C A210234 For a discussion and guide to related arrays, see A208510.
%F A210234 u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F A210234 v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
%F A210234 where u(1,x)=1, v(1,x)=1.
%e A210234 First five rows:
%e A210234 1
%e A210234 2...3
%e A210234 3...7....7
%e A210234 4...14...20...15
%e A210234 5...22...50...53...31
%e A210234 First three polynomials v(n,x): 1, 2 + 3x , 3 + 7x + 7x^2.
%t A210234 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210234 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A210234 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A210234 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210234 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210234 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210234 TableForm[cu]
%t A210234 Flatten[%]      (* A210233 *)
%t A210234 Table[Expand[v[n, x]], {n, 1, z}]
%t A210234 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210234 TableForm[cv]
%t A210234 Flatten[%]      (* A210234 *)
%Y A210234 Cf. A210233, A208510.
%K A210234 nonn,tabl
%O A210234 1,2
%A A210234 _Clark Kimberling_, Mar 20 2012