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 A157335 Expansion of 1/( (1+x)*(1-7*x+x^2) ). 2

%I

%S 1,6,42,287,1968,13488,92449,633654,4343130,29768255,204034656,

%T 1398474336,9585285697,65698525542,450304393098,3086432226143,

%U 21154721189904,144996616103184,993821591532385,6811754524623510,46688460080832186,320007466041201791

%N Expansion of 1/( (1+x)*(1-7*x+x^2) ).

%H Colin Barker, <a href="/A157335/b157335.txt">Table of n, a(n) for n = 0..1000</a>

%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry4/barry64.html">Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays</a>, JIS 12 (2009) 09.8.6

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,6,-1).

%F a(n) = A152119(n+1) * A152119(n+2).

%F a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*U(k,7/2) where U(n,x) is the n-th-Chebyshev polynomial of the second kind;

%F a(n) = Sum_{k=0..2n} F(n+1-floor(k/2))*F(n+1-mod(k,2)-floor(k/2))*A152119(k+1).

%F a(n) = 7*a(n-1) - a(n-2) + (-1)^n, n > 1. - _Vincenzo Librandi_, Mar 13 2011

%F 9*a(n) = 8*A004187(n+1) - A004187(n) + (-1)^n. - _R. J. Mathar_, Mar 15 2011

%F a(n)= ((-1)^(n+1)/3)*Sum_{k=0..n+1} (-1)^k*Fibonacci(4*k). - _Gary Detlefs_, Jan 21 2013

%F a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3), with a(0)=1, a(1)=6, a(2)=42. - _Harvey P. Dale_, Oct 25 2014

%F a(n) = (2^(-n)*(5*(-2)^n + (20-9*sqrt(5))*(7-3*sqrt(5))^n + (7+3*sqrt(5))^n*(20+9*sqrt(5))))/45. - _Colin Barker_, Mar 05 2016

%t CoefficientList[Series[1/((1+x)(1-7x+x^2)),{x,0,20}],x] (* or *) LinearRecurrence[{6,6,-1},{1,6,42},20] (* _Harvey P. Dale_, Oct 25 2014 *)

%o (PARI) Vec(1/(x^3-6*x^2-6*x+1) + O(x^30)) \\ _Colin Barker_, Mar 05 2016

%o (MAGMA) I:=[0, 6, 42]; [n le 3 select I[n] else 6*Self(n-1) + 6*Self(n-2) - Self(n-3): n in [1..30]]; // _G. C. Greubel_, Feb 01 2018

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 27 2009

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Last modified March 20 13:18 EDT 2019. Contains 321345 sequences. (Running on oeis4.)