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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).

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

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

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;

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G. C. Greubel, <a href="/A159764/b159764.txt">Rows n=0..100 of triangle, flattened</a>

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 *)

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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

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

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

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

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

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(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

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