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

1, -3, 1, 8, -6, 1, -21, 25, -9, 1, 55, -90, 51, -12, 1, -144, 300, -234, 86, -15, 1, 377, -954, 951, -480, 130, -18, 1, -987, 2939, -3573, 2305, -855, 183, -21, 1, 2584, -8850, 12707, -10008, 4740, -1386, 245, -24, 1, -6765, 26195, -43398, 40426, -23373, 8715

Offset: 0

Author

Philippe Deléham, Feb 20 2012

Keywords

• easy
• sign
• tabl

Comments

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

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.

Diagonal sums are (-3)^n.

Inverse array is A091965.

Examples

			Triangle begins:
      1;
     -3,     1;
      8,    -6,      1;
    -21,    25,     -9,      1;
     55,   -90,     51,    -12,      1;
   -144,   300,   -234,     86,    -15,     1;
    377,  -954,    951,   -480,    130,   -18,     1;
   -987,  2939,  -3573,   2305,   -855,   183,   -21,   1;
   2584, -8850,  12707, -10008,   4740, -1386,   245, -24,   1;
  -6765, 26195, -43398,  40426, -23373,  8715, -2100, 316, -27, 1;
Triangle (0, -3, 1/3, -1/3, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,    1;
  0,   -3,   1;
  0,    8,  -6,    1;
  0,  -21,  25,   -9,   1;
  0,   55, -90,   51, -12,   1;
  0, -144, 300, -234,  86, -15, 1;
  ...
		

Crossrefs

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).

Programs

Mathematica

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];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 22 2018 *)

PARI

row(n) = Vecrev(subst(polchebyshev(n,2,x/2), x, x-3))
tabf(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Jun 22 2018

Sage

@CachedFunction
def A207815(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A207815(n-1,k-1)-A207815(n-2,k)-3*A207815(n-1,k)
for n in (0..9): [A207815(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

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

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

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

Extensions

T(8,0) corrected by Jean-François Alcover, Jun 22 2018