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

1, 3, 7, 1, 15, 5, 31, 17, 1, 63, 49, 7, 127, 129, 31, 1, 255, 321, 111, 9, 511, 769, 351, 49, 1, 1023, 1793, 1023, 209, 11, 2047, 4097, 2815, 769, 71, 1, 4095, 9217, 7423, 2561, 351, 13, 8191, 20481, 18943, 7937, 1471, 97, 1, 16383, 45057, 47103

Offset

1,2

Comments

Column 1: -1+2^n

Row sums: A048739

Alternating row sums: triangular numbers, A000217

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

Links

Table of n, a(n) for n=1..52.

K. Dilcher, K. B. Stolarsky, Nonlinear recurrences related to Chebyshev polynomials, The Ramanujan Journal, 2014, Online Oct. 2014, pp. 1-23. See Table 1.

Formula

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

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

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

Example

First five rows:
1
3
7....1
15...5
31...17...1
First three polynomials u(n,x): 1, 3, 7 + x.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210197 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210198 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A048739 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A005409 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000217 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)

Crossrefs

Cf. A210198, A208510.

Essentially the same as the triangle in A257597.

Sequence in context: A135561 A196231 A210037 * A324715 A316665 A110238

Adjacent sequences: A210194 A210195 A210196 * A210198 A210199 A210200

Keyword

nonn,tabf

Author

Clark Kimberling, Mar 18 2012

Status

approved