A257597 -id:A257597 - cp's OEIS frontend

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

Clark Kimberling, Mar 18 2012

Column 1:  -1+2^n
Row sums:  A048739
Alternating row sums: triangular numbers, A000217
For a discussion and guide to related arrays, see A208510.

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

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

Cf. A210198, A208510.

Essentially the same as the triangle in A257597.

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 *)

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.

A257598 Triangle read by rows: coefficients of polynomials W_n(x), highest degree terms first.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 16, -4, 1, 1, 1, 1, 32, -16, 2, 1, 1, 1, 1, 64, -48, 8, 1, 1, 1, 1, 1, 128, -128, 32, 1, 1, 1, 1, 1, 256, -320, 112, -8, 1, 1, 1, 1, 1, 1, 512, -768, 352, -48, 2, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Jun 06 2015

			Triangle of coefficients begins:
1,
1, 1,
2, 1, 1,
4, 1, 1, 1,
8, 1, 1, 1,
16, -4, 1, 1, 1, 1,
32, -16, 2, 1, 1, 1, 1,
64, -48, 8, 1, 1, 1, 1, 1,
128, -128, 32, 1, 1, 1, 1, 1,
256, -320, 112 -8, 1, 1, 1, 1, 1, 1,
512, -768, 352 -48, 2, 1, 1, 1, 1, 1, 1,
...
The actual polynomials are:
0 1
1 x^2 + 1
2 2x^4 + x^2 + 1
3 4x^6 + x^4 + x^2 + 1
4 8x^8 + x^4 + x^2 + 1
5 16x^10 - 4x^8 + x^6 + x^4 + x^2 + 1
6 32x^12 - 16x^10 + 2x^8 + x^6 + x^4 + x^2 + 1
7 64x^14 - 48x^12 + 8x^10 + x^8 + x^6 + x^4 + x^2 + 1
8 128x^16 - 128x^14 + 32x^12 + x^8 + x^6 + x^4 + x^2 + 1
9 256x^18 - 320x^16 + 112x^14 - 8x^12 + x^10 + x^8 + x^6 + x^4 + x^2 + 1
10 512x^20 - 768x^18 + 352x^16 - 48x^14 + 2x^12 + x^10 + x^8 + x^6 + x^4 + x^2 + 1
...

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

Cf. A257597.

tabf(nn) = {pp = 1; p = x; for (n=1, nn, np = 2*x*p-pp-x^(n+1); w = p^2 - pp*np; forstep (j=poldegree(w), 0, -1, if (c = polcoeff(w, j), print1(c, ", "));); pp = p; p = np; print(););} \\ Michel Marcus, Aug 22 2015

W(n) = V(n+1)^2 - V(n)*V(n+2) where V(n) are the polynomials defined in A257597. - Michel Marcus, Aug 22 2015

One typo in data corrected by Michel Marcus, Aug 22 2015

cp's OEIS Frontend

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

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