A136159 A Chebyshev polynomial triangle of the first kind defined by T(n+1,x) = 3x*T(n,x) - T(n-1,x).

1, 1, 3, -1, 9, -4, 27, -15, 1, 81, -54, 7, 243, -189, 36, -1, 729, -648, 162, -10, 2187, -2187, 675, -66, 1, 6561, -7290, 2673, -360, 13, 19683, -24057, 10206, -1755, 105, -1, 59049, -78732, 37908, -7938, 675, -16

Offset: 0

Author

Gary W. Adamson, Dec 16 2007

Keywords

• tabf
• sign

Comments

Row sums (unsigned) give A003688, (starting 1, 1, 4, 13, 43, 142, 469, ...).

Examples

			First few rows of the polynomials are:
1;
x;
3x^2 - 1;
9x^3 - 4x;
27x^4 - 15x^2 + 1;
81x^5 - 54x^3 + 7x;
243x^6 - 189x^4 + 36x^2 - 1;
729x^7 - 648x^5 + 162x^3 - 10x;
...
		

Crossrefs

Cf. A136158, A003688.

Cf. A000244, A006234, A080419, A080420, A080421, A080422, A080423. [Philippe Deléham, Sep 12 2009]

Programs

PARI

P(n) = if (n==0, 1, if (n==1, x, 3*x*P(n-1) - P(n-2)));
row(n) = select(x->x!=0, Vec(P(n))); \\ Michel Marcus, Apr 15 2018

Formula

T(0,x) = 1, T(1,x) = x, T(n+1,x) = 3x*T(n,x) - T(n-1,x).

G.f: (l - tx)/(1 - 3tx + t^2).

Given triangle A136158, shift down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row.

Extensions

Corrected and extended by Philippe Deléham, Sep 12 2009

Keyword tabf set by Michel Marcus, Apr 15 2018