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

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

Author

N. J. A. Sloane, Jun 06 2015

Keywords

• sign
• tabl

Examples

			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
...
		

Links

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

Crossrefs

Cf. A257597.

Programs

PARI

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

Formula

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

Extensions

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