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Triangle of coefficients of Chebyshev's S(n,x-3) polynomials (exponents of x in increasing order).
8

%I #23 Jun 22 2018 23:26:31

%S 1,-3,1,8,-6,1,-21,25,-9,1,55,-90,51,-12,1,-144,300,-234,86,-15,1,377,

%T -954,951,-480,130,-18,1,-987,2939,-3573,2305,-855,183,-21,1,2584,

%U -8850,12707,-10008,4740,-1386,245,-24,1,-6765,26195,-43398,40426,-23373,8715

%N Triangle of coefficients of Chebyshev's S(n,x-3) polynomials (exponents of x in increasing order).

%C Riordan array (1/(1+3*x+x^2), x/(1+3*x+x^2)).

%C Subtriangle of the triangle given by (0, -3, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%C Diagonal sums are (-3)^n.

%C Inverse array is A091965.

%F T(n,k) = (-1)^(n-k)*A125662(n,k).

%F Recurrence: T(n,k) = (-3)*T(n-1,k) + T(n-1,k-1) - T(n-2,k).

%F G.f.: 1/(1+3*x+x^2-y*x).

%e Triangle begins:

%e 1;

%e -3, 1;

%e 8, -6, 1;

%e -21, 25, -9, 1;

%e 55, -90, 51, -12, 1;

%e -144, 300, -234, 86, -15, 1;

%e 377, -954, 951, -480, 130, -18, 1;

%e -987, 2939, -3573, 2305, -855, 183, -21, 1;

%e 2584, -8850, 12707, -10008, 4740, -1386, 245, -24, 1;

%e -6765, 26195, -43398, 40426, -23373, 8715, -2100, 316, -27, 1;

%e Triangle (0, -3, 1/3, -1/3, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:

%e 1;

%e 0, 1;

%e 0, -3, 1;

%e 0, 8, -6, 1;

%e 0, -21, 25, -9, 1;

%e 0, 55, -90, 51, -12, 1;

%e 0, -144, 300, -234, 86, -15, 1;

%e ...

%t T[_?Negative, _] = 0; T[0, 0] = 1; T[0, _] = 0; T[n_, n_] = 1; T[n_, k_] := T[n, k] = T[n - 1, k - 1] - T[n - 2, k] - 3 T[n - 1, k];

%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* _Jean-François Alcover_, Jun 22 2018 *)

%o (Sage)

%o @CachedFunction

%o def A207815(n,k):

%o if n< 0: return 0

%o if n==0: return 1 if k == 0 else 0

%o return A207815(n-1,k-1)-A207815(n-2,k)-3*A207815(n-1,k)

%o for n in (0..9): [A207815(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012

%o (PARI) row(n) = Vecrev(subst(polchebyshev(n,2,x/2), x, x-3))

%o tabf(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Jun 22 2018

%Y Cf. Chebyshev's S(n,x+k) polynomials: A207824 (k = 5), A207823 (k = 4), A125662 (k = 3), A078812 (k = 2), A101950 (k = 1), A049310 (k = 0), A104562 (k = -1), A053122 (k = -2), A207815 (k = -3), A159764 (k = -4), A123967 (k = -5).

%K easy,sign,tabl

%O 0,2

%A _Philippe Deléham_, Feb 20 2012

%E T(8,0) corrected by _Jean-François Alcover_, Jun 22 2018