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A207815 Triangle of coefficients of Chebyshev's S(n,x-3) polynomials (exponents of x in increasing order).

%I A207815 #23 Jun 22 2018 23:26:31
%S A207815 1,-3,1,8,-6,1,-21,25,-9,1,55,-90,51,-12,1,-144,300,-234,86,-15,1,377,
%T A207815 -954,951,-480,130,-18,1,-987,2939,-3573,2305,-855,183,-21,1,2584,
%U A207815 -8850,12707,-10008,4740,-1386,245,-24,1,-6765,26195,-43398,40426,-23373,8715
%N A207815 Triangle of coefficients of Chebyshev's S(n,x-3) polynomials (exponents of x in increasing order).
%C A207815 Riordan array (1/(1+3*x+x^2), x/(1+3*x+x^2)).
%C A207815 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 A207815 Diagonal sums are (-3)^n.
%C A207815 Inverse array is A091965.
%F A207815 T(n,k) = (-1)^(n-k)*A125662(n,k).
%F A207815 Recurrence: T(n,k) = (-3)*T(n-1,k) + T(n-1,k-1) - T(n-2,k).
%F A207815 G.f.: 1/(1+3*x+x^2-y*x).
%e A207815 Triangle begins:
%e A207815       1;
%e A207815      -3,     1;
%e A207815       8,    -6,      1;
%e A207815     -21,    25,     -9,      1;
%e A207815      55,   -90,     51,    -12,      1;
%e A207815    -144,   300,   -234,     86,    -15,     1;
%e A207815     377,  -954,    951,   -480,    130,   -18,     1;
%e A207815    -987,  2939,  -3573,   2305,   -855,   183,   -21,   1;
%e A207815    2584, -8850,  12707, -10008,   4740, -1386,   245, -24,   1;
%e A207815   -6765, 26195, -43398,  40426, -23373,  8715, -2100, 316, -27, 1;
%e A207815 Triangle (0, -3, 1/3, -1/3, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
%e A207815   1;
%e A207815   0,    1;
%e A207815   0,   -3,   1;
%e A207815   0,    8,  -6,    1;
%e A207815   0,  -21,  25,   -9,   1;
%e A207815   0,   55, -90,   51, -12,   1;
%e A207815   0, -144, 300, -234,  86, -15, 1;
%e A207815   ...
%t A207815 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 A207815 Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* _Jean-François Alcover_, Jun 22 2018 *)
%o A207815 (Sage)
%o A207815 @CachedFunction
%o A207815 def A207815(n,k):
%o A207815     if n< 0: return 0
%o A207815     if n==0: return 1 if k == 0 else 0
%o A207815     return A207815(n-1,k-1)-A207815(n-2,k)-3*A207815(n-1,k)
%o A207815 for n in (0..9): [A207815(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%o A207815 (PARI) row(n) = Vecrev(subst(polchebyshev(n,2,x/2), x, x-3))
%o A207815 tabf(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Jun 22 2018
%Y A207815 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 A207815 easy,sign,tabl
%O A207815 0,2
%A A207815 _Philippe Deléham_, Feb 20 2012
%E A207815 T(8,0) corrected by _Jean-François Alcover_, Jun 22 2018