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%I #27 Sep 08 2022 08:45:24
%S 1,1,2,6,15,35,84,204,493,1189,2870,6930,16731,40391,97512,235416,
%T 568345,1372105,3312554,7997214,19306983,46611179,112529340,271669860,
%U 655869061,1583407981,3822685022,9228778026,22280241075,53789260175
%N Expansion of (1-x)/((1-x)^2 - x^2*(1+x)^2).
%C Diagonal sums of number triangle A114123.
%C Binomial transform of A114122.
%C Congruent to 1,1,0,0,1,1,... modulo 2.
%H Vincenzo Librandi, <a href="/A116404/b116404.txt">Table of n, a(n) for n = 0..1000</a>
%H Vincent Vatter, <a href="https://arxiv.org/abs/1605.04297">Growth rates of permutation classes: from countable to uncountable</a>, arXiv:1605.04297 [math.CO], 2016.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,2,1).
%F G.f.: (1-x)/(1 - 2*x - 2*x^3 - x^4).
%F a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4).
%F a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(2*k,n-2*k-j)*C(n-2*k,j)*2^(n-2*k-j).
%F 2*a(n) = A056594(n) + A000129(n+1). - _R. J. Mathar_, Oct 25 2011
%F a(n) = Sum_{k=0..floor(n/2)} hypergeometric2F1([-2*k, -n+2*k], [1], 2). - _G. C. Greubel_, Nov 20 2021
%t LinearRecurrence[{2,0,2,1}, {1,1,2,6}, 35] (* _Emanuele Munarini_, Apr 27 2017 *)
%t CoefficientList[Series[(1-x)/((1-x)^2 -x^2(1+x)^2), {x, 0, 35}], x] (* _Vincenzo Librandi_, Aug 19 2017 *)
%o (PARI) Vec((1-x)/((1-x)^2-x^2*(1+x)^2) + O(x^40)) \\ _Michel Marcus_, Aug 19 2017
%o (Magma) I:=[1,1,2,6]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-3)+Self(n-4): n in [1..36]]; // _Vincenzo Librandi_, Aug 19 2017
%o (Sage)
%o def A116404(n): return sum( round( hypergeometric([-n+2*k, -2*k], [1], 2) ) for k in (0..n//2) )
%o [A116404(n) for n in (0..35)] # _G. C. Greubel_, Nov 20 2021
%Y Cf. A000129, A056594, A114122, A114123.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Feb 07 2006
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