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%I #35 Jul 22 2024 14:07:59
%S 1,2,5,14,37,98,261,694,1845,4906,13045,34686,92229,245234,652069,
%T 1733830,4610197,12258362,32594581,86667918,230447141,612751362,
%U 1629285701,4332217046,11519222517,30629233482,81442123573,216551925662,575804441861,1531045056530
%N Expansion of 1/(1-2*x-x^2-2*x^3).
%C Number of compositions of n into two sorts of parts 1 and 3, and one sort of parts 2. - _Joerg Arndt_, May 02 2015
%H Michael De Vlieger, <a href="/A077938/b077938.txt">Table of n, a(n) for n = 0..2354</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1804.05027">The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1804.05027 [math.CO], 2018.
%H Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, <a href="http://www.emis.de/journals/INTEGERS/papers/p16/p16.pdf">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (<a href="http://arxiv.org/abs/1302.2274">arXiv:1302.2274</a>)
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,2)
%F a(n)=sum(k=1..n, sum(j=0..k, binomial(j,n-3*k+2*j)* binomial(k,j)*2^(-n+4*k-2*j))), n>0, a(0)=1. [_Vladimir Kruchinin_, May 05 2011]
%F a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3). [_Stefan Schuster_, Apr 24 2015]
%t LinearRecurrence[{2, 1, 2}, {1, 2, 5}, 100] (* _Vladimir Joseph Stephan Orlovsky_, Jul 03 2011 *)
%t CoefficientList[Series[1/(1-2x-x^2-2x^3),{x,0,30}],x] (* _Harvey P. Dale_, Jul 22 2024 *)
%o (Maxima) a(n):=sum(sum(binomial(j,n-3*k+2*j)*binomial(k,j)*2^(-n+4*k-2*j),j,0,k),k,1,n); // _Vladimir Kruchinin_, May 05 2011
%o (PARI) Vec(1/(1-2*x-x^2-2*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012
%Y Absolute values of A077987.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 17 2002
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