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A210197 Triangle of coefficients of polynomials u(n,x) jointly generated with A210198; see the Formula section. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 20 04:37 EDT 2019. Contains 328247 sequences. (Running on oeis4.)