A136159 - OEIS

%I #20 Apr 15 2018 11:11:34

%S 1,1,3,-1,9,-4,27,-15,1,81,-54,7,243,-189,36,-1,729,-648,162,-10,2187,

%T -2187,675,-66,1,6561,-7290,2673,-360,13,19683,-24057,10206,-1755,105,

%U -1,59049,-78732,37908,-7938,675,-16

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

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

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

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

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

%e First few rows of the polynomials are:

%e 1;

%e x;

%e 3x^2 - 1;

%e 9x^3 - 4x;

%e 27x^4 - 15x^2 + 1;

%e 81x^5 - 54x^3 + 7x;

%e 243x^6 - 189x^4 + 36x^2 - 1;

%e 729x^7 - 648x^5 + 162x^3 - 10x;

%e ...

%o (PARI) P(n) = if (n==0, 1, if (n==1, x, 3*x*P(n-1) - P(n-2)));

%o row(n) = select(x->x!=0, Vec(P(n))); \\ _Michel Marcus_, Apr 15 2018

%Y Cf. A136158, A003688.

%Y Cf. A000244, A006234, A080419, A080420, A080421, A080422, A080423. [_Philippe Deléham_, Sep 12 2009]

%K tabf,sign

%O 0,3

%A _Gary W. Adamson_, Dec 16 2007

%E Corrected and extended by _Philippe Deléham_, Sep 12 2009

%E Keyword tabf set by _Michel Marcus_, Apr 15 2018