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%I #29 Sep 08 2022 08:45:48
%S 1,-1,-11,-50,-186,-631,-2029,-6299,-19075,-56704,-166164,-481391,
%T -1381691,-3935125,-11134331,-31328366,-87721614,-244588519,
%U -679429225,-1881102959,-5192705779,-14296088956,-39263958696,-107601905375,-294291714551,-803416991401
%N Hankel transform of a simple Catalan convolution.
%C Hankel transform of A167422.
%H G. C. Greubel, <a href="/A167423/b167423.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).
%F G.f.: ( 1-7*x+6*x^2-x^3 ) / (x^2-3*x+1)^2 .
%F a(n) = F(2*n)*(1-3*n)/2 + L(2*n)*(1-n)/2. - _Paul Barry_, Feb 22 2010
%F a(n) = 3*A001871(n-1) - 2*A001871(n) + F(2*n+4). - _Ralf Stephan_, May 21 2014
%F a(n) = 1 - Sum_{k=1..n} k*F(2*k+1), where F(n) = A000045(n). - _Vladimir Reshetnikov_, Oct 28 2015
%t Table[((1-3n) Fibonacci[2n] + (1-n) LucasL[2n])/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 28 2015 *)
%t LinearRecurrence[{6, -11, 6, -1}, {1, -1, -11, -50}, 50] (* _G. C. Greubel_, Jun 12 2016 *)
%o (PARI) Vec((1-7*x+6*x^2-x^3)/(1-6*x+11*x^2-6*x^3+x^4) + O(x^100)) \\ _Altug Alkan_, Oct 29 2015
%o (Magma) [Fibonacci(2*n)*(1-3*n)/2 + Lucas(2*n)*(1-n)/2: n in [0..30]]; // _Vincenzo Librandi_, Jun 13 2016
%Y Cf. A167422, A000045, A000032.
%K easy,sign
%O 0,3
%A _Paul Barry_, Nov 03 2009
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