A159764 Riordan array (1/(1+4x+x^2), x/(1+4x+x^2)).

1, -4, 1, 15, -8, 1, -56, 46, -12, 1, 209, -232, 93, -16, 1, -780, 1091, -592, 156, -20, 1, 2911, -4912, 3366, -1200, 235, -24, 1, -10864, 21468, -17784, 8010, -2120, 330, -28, 1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, -151316, 386373

Offset: 0

Author

Paul Barry, Apr 21 2009

Keywords

• easy
• sign
• tabl

Comments

Row sums are (-1)^n*F(2n+2). Diagonal sums are (-1)^n*4^n. Inverse is A052179.

The positive matrix is (1/(1-4x+x^2), x/(1-4x+x^2)) with general term T(n,k) = if(k<=n, Gegenbauer_C(n-k,k+1,2),0).

For another version, see A124029.

Triangle of coefficients of Chebyshev's S(n,x-4) polynomials (exponents of x in increasing order). - Philippe Deléham, Feb 22 2012

Subtriangle of triangle given by (0, -4, 1/4, -1/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 22 2012

Examples

			Triangle begins
     1;
    -4,     1;
    15,    -8,     1;
   -56,    46,   -12,     1;
   209,  -232,    93,   -16,     1;
  -780,  1091,  -592,   156,   -20,     1;
  2911, -4912,  3366, -1200,   235,   -24,     1;
Triangle (0, -4, 1/4, -1/4, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,    1;
  0,   -4,    1;
  0,   15,   -8,    1;
  0,  -56,   46,  -12,    1;
  0,  209, -232,   93,  -16,    1;
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened

Crossrefs

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

Programs

Mathematica

CoefficientList[CoefficientList[Series[1/(1 + 4*x + x^2 - y*x), {x, 0, 10}, {y, 0, 10}], x], y]//Flatten (* G. C. Greubel, May 21 2018 *)

Sage

@CachedFunction
def A159764(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A159764(n-1,k-1)-A159764(n-2,k)-4*A159764(n-1,k)
for n in (0..9): [A159764(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

Number triangle T(n,k) = if(k<=n, Gegenbauer_C(n-k,k+1,-2),0).

G.f.: 1/(1+4*x+x^2-y*x). - Philippe Deléham, Feb 22 2012

T(n,k) = (-4)*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Feb 22 2012