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A052544 Expansion of (1-x)^2/(1 - 4*x + 3*x^2 - x^3). 7

%I #97 Sep 08 2022 08:44:59

%S 1,2,6,19,60,189,595,1873,5896,18560,58425,183916,578949,1822473,

%T 5736961,18059374,56849086,178955183,563332848,1773314929,5582216355,

%U 17572253481,55315679788,174128175064,548137914373,1725482812088

%N Expansion of (1-x)^2/(1 - 4*x + 3*x^2 - x^3).

%C Equals INVERT transform of (1, 1, 3, 8, 21, 55, 144, ...). - _Gary W. Adamson_, May 01 2009

%C The Ze2 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence. - _Johannes W. Meijer_, Aug 16 2011

%C Equals the partial sums of A052529 starting (1, 1, 4, 13, 41, 129, ...). - _Gary W. Adamson_, Feb 15 2012

%C First trisection of Narayana's cows sequence A000930. - _Oboifeng Dira_, Aug 03 2016

%C From _Peter Bala_, Nov 03 2017: (Start)

%C Let f(x) = x/(1 - x^3), the characteristic function of numbers of the form 3*n + 1. Then f(f(x)) = Sum_{n >= 0} a(n)*x^(3*n+1).

%C a(n) = the number of compositions of 3*n + 1 into parts of the form 3*m + 1. For example, a(2) = 6 and the six compositions of 7 into parts of the form 3*m + 1 are 7, 4 + 1 + 1 + 1, 1 + 4 + 1 + 1, 1 + 1 + 4 + 1, 1 + 1 + 1 + 4 and 1 + 1 + 1 + 1 + 1 + 1 + 1. Cf. A001519, which gives the number of compositions of an odd number into odd parts. (End)

%C a(n-1) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 4>2} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements, and the fourth element is larger than the second element. - _Sergey Kitaev_, Dec 09 2020

%H Vincenzo Librandi, <a href="/A052544/b052544.txt">Table of n, a(n) for n = 0..500</a>

%H Alice L. L. Gao, Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.

%H Alice L. L. Gao, Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=480">Encyclopedia of Combinatorial Structures 480</a>

%H Sergey Kitaev and Artem Pyatkin, <a href="https://arxiv.org/abs/2204.08936">On permutations avoiding partially ordered patterns defined by bipartite graphs</a>, arXiv:2204.08936 [math.CO], 2022.

%H H. Stephan, <a href="http://www.wias-berlin.de/people/stephan/tdm02.1.pdf">Rekursive Folgen im Pascalschen Dreieck</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,1).

%F G.f.: (1-x)^2/(1 -4*x +3*x^2 -x^3).

%F a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3).

%F a(n) = Sum(-1/31*(-4-7*_alpha+2*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+4*_Z-3*_Z^2+_Z^3)).

%F a(n) = Sum_{k=0..n} binomial(n+2*k, 3*k). - _Richard L. Ollerton_, May 12 2004

%F G.f.: 1 / (1 - x - x / (1 - x)^2). - _Michael Somos_, Jan 12 2012

%F a(n) = hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], -4/27). - _Peter Luschny_, Nov 03 2017

%e G.f. = 1 + 2*x + 6*x^2 + 19*x^3 + 60*x^4 + 189*x^5 + 595*x^6 + 1873*x^7 + ...

%p spec := [S,{S=Sequence(Union(Z,Prod(Z,Sequence(Z),Sequence(Z))))}, unlabeled]: seq(combstruct[count](spec,size=n), n=0..25);

%p A052544 := proc(n): add(binomial(n+2*k, 3*k), k=0...n) end: seq(A052544(n), n=0..25); # _Johannes W. Meijer_, Aug 16 2011

%t LinearRecurrence[{4,-3,1},{1,2,6},30] (* _Harvey P. Dale_, Jul 13 2011 *)

%t Table[Sum[Binomial[n + 2 k, 3 k], {k, 0, n}], {n, 0, 30}] (* or *)

%t CoefficientList[Series[(1-x)^2/(1-4x+3x^2-x^3), {x, 0, 30}], x] (* _Michael De Vlieger_, Aug 03 2016 *)

%o (PARI) {a(n) = sum(k=0, n, binomial(n + 2*k, 3*k))}; /* _Michael Somos_, Jan 12 2012 */

%o (PARI) Vec((1-x)^2/(1-4*x+3*x^2-x^3)+O(x^30)) \\ _Charles R Greathouse IV_, Jan 12 2012

%o (Magma) I:=[1, 2, 6]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2) +Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jan 12 2012

%o (Sage) ((1-x)^2/(1-4*x+3*x^2-x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 09 2019

%o (GAP) a:=[1,2,6];; for n in [4..30] do a[n]:=4*a[n-1]-3*a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, May 09 2019

%Y Cf. A124820 (partial sums).

%Y Cf. A052529, A001519.

%K easy,nonn

%O 0,2

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E More terms from _James A. Sellers_, Jun 06 2000

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)