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

0,3

Comments

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

Links

Table of n, a(n) for n=0..41.

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.

Example

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;
...

Prog

(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

Crossrefs

Cf. A136158, A003688.

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

Sequence in context: A127550 A021317 A091579 * A005533 A331257 A112626

Adjacent sequences: A136156 A136157 A136158 * A136160 A136161 A136162

Keyword

tabf,sign

Author

Gary W. Adamson, Dec 16 2007

Extensions

Corrected and extended by Philippe Deléham, Sep 12 2009

Keyword tabf set by Michel Marcus, Apr 15 2018

Status

approved