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A210197 Triangle of coefficients of polynomials u(n,x) jointly generated with A210198; see the Formula section.

%I A210197 #12 Aug 22 2015 03:00:44
%S A210197 1,3,7,1,15,5,31,17,1,63,49,7,127,129,31,1,255,321,111,9,511,769,351,
%T A210197 49,1,1023,1793,1023,209,11,2047,4097,2815,769,71,1,4095,9217,7423,
%U A210197 2561,351,13,8191,20481,18943,7937,1471,97,1,16383,45057,47103
%N A210197 Triangle of coefficients of polynomials u(n,x) jointly generated with A210198; see the Formula section.
%C A210197 Column 1:  -1+2^n
%C A210197 Row sums:  A048739
%C A210197 Alternating row sums: triangular numbers, A000217
%C A210197 For a discussion and guide to related arrays, see A208510.
%H A210197 K. Dilcher, K. B. Stolarsky, <a href="http://dx.doi.org/10.1007/s11139-014-9620-5">Nonlinear recurrences related to Chebyshev polynomials</a>, The Ramanujan Journal, 2014, Online Oct. 2014, pp. 1-23. See Table 1.
%F A210197 u(n,x)=u(n-1,x)+v(n-1,x)+1,
%F A210197 v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
%F A210197 where u(1,x)=1, v(1,x)=1.
%e A210197 First five rows:
%e A210197 1
%e A210197 3
%e A210197 7....1
%e A210197 15...5
%e A210197 31...17...1
%e A210197 First three polynomials u(n,x): 1, 3, 7 + x.
%t A210197 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210197 u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A210197 v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
%t A210197 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210197 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210197 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210197 TableForm[cu]
%t A210197 Flatten[%]   (* A210197 *)
%t A210197 Table[Expand[v[n, x]], {n, 1, z}]
%t A210197 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210197 TableForm[cv]
%t A210197 Flatten[%]  (* A210198 *)
%t A210197 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A048739 *)
%t A210197 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A005409 *)
%t A210197 Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000217 *)
%t A210197 Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)
%Y A210197 Cf. A210198, A208510.
%Y A210197 Essentially the same as the triangle in A257597.
%K A210197 nonn,tabf
%O A210197 1,2
%A A210197 _Clark Kimberling_, Mar 18 2012