A210197 - OEIS

%I #12 Aug 22 2015 03:00:44

%S 1,3,7,1,15,5,31,17,1,63,49,7,127,129,31,1,255,321,111,9,511,769,351,

%T 49,1,1023,1793,1023,209,11,2047,4097,2815,769,71,1,4095,9217,7423,

%U 2561,351,13,8191,20481,18943,7937,1471,97,1,16383,45057,47103

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

%C Column 1:  -1+2^n

%C Row sums:  A048739

%C Alternating row sums: triangular numbers, A000217

%C For a discussion and guide to related arrays, see A208510.

%H 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 u(n,x)=u(n-1,x)+v(n-1,x)+1,

%F v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 3

%e 7....1

%e 15...5

%e 31...17...1

%e First three polynomials u(n,x): 1, 3, 7 + x.

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;

%t v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]   (* A210197 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]  (* A210198 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A048739 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A005409 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000217 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)

%Y Cf. A210198, A208510.

%Y Essentially the same as the triangle in A257597.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Mar 18 2012