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%I #98 Oct 12 2022 04:30:36
%S 1,1,4,13,41,129,406,1278,4023,12664,39865,125491,395033,1243524,
%T 3914488,12322413,38789712,122106097,384377665,1209982081,3808901426,
%U 11990037126,37743426307,118812495276,374009739309,1177344897715
%N Expansion of (1-x)^3/(1 - 4*x + 3*x^2 - x^3).
%C a(n+1) is the number of distinct matrix products in (A+B+C+D)^n where commutator [A,B] = [A,C] = [B,C] = 0 but D does not commute with A, B, or C. - _Paul D. Hanna_ and _Max Alekseyev_, Feb 01 2006
%C Starting (1, 4, 13, ...) = INVERT transform of the triangular series, (1, 3, 6, 10, ...). Example: a(5) = 129 = termwise products of (1, 1, 4, 13, 41) and (15, 10, 6, 3, 1) = (15 + 10 + 24 + 39 + 41). - _Gary W. Adamson_, Apr 10 2009
%C a(n) is the number of generalized compositions of n when there are i^2/2+i/2 different types of i, (i=1,2,...). - _Milan Janjic_, Sep 24 2010
%C Dedrickson (Section 4.2) gives a bijection between colored compositions of n, where each part k has one of binomial(k+1,2) colors, and 0,1,2,3 strings of length n-1 avoiding 10, 20 and 21. Cf. A095263. For a refinement of this sequence counting binomial(k+1,2)-colored compositions by the number of parts see A127893. - _Peter Bala_, Sep 17 2013
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 80.
%H Vincenzo Librandi, <a href="/A052529/b052529.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3 , example 14.
%H C. R. Dedrickson III, <a href="https://digitalcommons.georgiasouthern.edu/etd/17">Compositions, Bijections, and Enumerations</a> Thesis, Jack N. Averitt College of Graduate Studies, Georgia Southern University, 2012.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=459">Encyclopedia of Combinatorial Structures 459</a>
%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,1).
%F a(n) = Sum_{a=0..n} (Sum_{b=0..n} (Sum_{c=0..n} C(n-b-c,a)*C(n-a-c,b)*C(n-a-b,c))).
%F G.f.: (1 - x)^3/(1 - 4*x + 3*x^2 - x^3).
%F a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3) for n>=4.
%F a(n) = Sum_{alpha = RootOf(-1+4*x-3*x^2+x^3)} (1/31)*(6 - 5*alpha - 3*alpha^2) * alpha^(-1-n).
%F For n>0, a(n) = Sum_{k=0..n-1} Sum_{i=0..k} Sum_{j=0..i} a(j). - _Benoit Cloitre_, Jan 26 2003
%F a(n) = Sum_{k=0..n} binomial(n+2*k-1, n-k). - _Vladeta Jovovic_, Mar 23 2003
%F If p[i]=i(i+1)/2 and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - _Milan Janjic_, May 02 2010
%F Recurrence equation: a(n) = Sum_{k = 1..n} 1/2*k*(k+1)*a(n-k) with a(0) = 1. - _Peter Bala_, Sep 19 2013
%F a(n) = Sum_{i=0..n} (n-i)*A052544(i) = A052544(n) - A052544(n-1) for n>=1. - _Areebah Mahdia_, Jul 07 2020
%p spec := [S,{S=Sequence(Prod(Z,Sequence(Z),Sequence(Z),Sequence(Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p f:= gfun:-rectoproc({a(n+4)-4*a(n+3)+3*a(n+2)-a(n+1), a(0) = 1, a(1) = 1, a(2) = 4, a(3) = 13},a(n),`remember`):
%p seq(f(n),n=0..40); # _Robert Israel_, Dec 19 2014
%t CoefficientList[Series[(-1+x)^3/(-1+4*x-3*x^2+x^3),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 22 2012 *)
%t LinearRecurrence[{4,-3,1},{1,1,4,13},30] (* _Harvey P. Dale_, Oct 04 2015 *)
%o (Magma) I:=[1, 1, 4, 13, 41, 129]; [n le 6 select I[n] else 4*Self(n-1) -3*Self(n-2)+Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jun 22 2012
%o (PARI) my(x='x+O('x^30)); Vec((1-x)^3/(1-4*x+3*x^2-x^3)) \\ _G. C. Greubel_, May 12 2019
%o (Sage) ((1-x)^3/(1-4*x+3*x^2-x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 12 2019
%Y Cf. A001906, A052530, A055991, A095263.
%Y Trisection of A000930.
%Y First differences of A052544.
%Y Row sums of triangle A127893.
%K nonn,easy
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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