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A159764 Riordan array (1/(1+4x+x^2), x/(1+4x+x^2)). 13

%I

%S 1,-4,1,15,-8,1,-56,46,-12,1,209,-232,93,-16,1,-780,1091,-592,156,-20,

%T 1,2911,-4912,3366,-1200,235,-24,1,-10864,21468,-17784,8010,-2120,330,

%U -28,1,40545,-91824,89238,-48624,16255,-3416,441,-32,1,-151316,386373

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

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

%C 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).

%C For another version, see A124029.

%C Triangle of coefficients of Chebyshev's S(n,x-4) polynomials (exponents of x in increasing order). - _Philippe Deléham_, Feb 22 2012

%C 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

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

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

%F G.f.: 1/(1+4*x+x^2-y*x). - _Philippe Deléham_, Feb 22 2012

%F T(n,k) = (-4)*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - _Philippe Deléham_, Feb 22 2012

%e Triangle begins

%e 1;

%e -4, 1;

%e 15, -8, 1;

%e -56, 46, -12, 1;

%e 209, -232, 93, -16, 1;

%e -780, 1091, -592, 156, -20, 1;

%e 2911, -4912, 3366, -1200, 235, -24, 1;

%e Triangle (0, -4, 1/4, -1/4, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:

%e 1;

%e 0, 1;

%e 0, -4, 1;

%e 0, 15, -8, 1;

%e 0, -56, 46, -12, 1;

%e 0, 209, -232, 93, -16, 1;

%t 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 *)

%o (Sage)

%o @CachedFunction

%o def A159764(n,k):

%o if n< 0: return 0

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

%o return A159764(n-1,k-1)-A159764(n-2,k)-4*A159764(n-1,k)

%o for n in (0..9): [A159764(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012

%Y 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.

%K easy,sign,tabl

%O 0,2

%A _Paul Barry_, Apr 21 2009

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Last modified December 15 11:43 EST 2019. Contains 329999 sequences. (Running on oeis4.)