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 A052529 Expansion of (1-x)^3/(1 - 4*x + 3*x^2 - x^3). 20
 1, 1, 4, 13, 41, 129, 406, 1278, 4023, 12664, 39865, 125491, 395033, 1243524, 3914488, 12322413, 38789712, 122106097, 384377665, 1209982081, 3808901426, 11990037126, 37743426307, 118812495276, 374009739309, 1177344897715 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 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 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 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 REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 80. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 D. Birmajer, J. B. Gil, M. D. Weiner, n the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 , example 14. C. R. Dedrickson III, Compositions, Bijections, and Enumerations Thesis, Jack N. Averitt College of Graduate Studies, Georgia Southern University, 2012. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 459 Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2. Index entries for linear recurrences with constant coefficients, signature (4,-3,1). FORMULA 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))). G.f.: (1 - x)^3/(1 - 4*x + 3*x^2 - x^3). a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3) for n>=4. a(n) = Sum_{alpha = RootOf(-1+4*x-3*x^2+x^3)} (1/31)*(6 - 5*alpha - 3*alpha^2) * alpha^(-1-n). 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 a(n) = Sum_{k=0..n} binomial(n+2*k-1, n-k). - Vladeta Jovovic, Mar 23 2003 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 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 a(n) = Sum_{i=0..n} (n-i)*A052544(i) = A052544(n) - A052544(n-1) for n>=1. - Areebah Mahdia, Jul 07 2020 MAPLE spec := [S, {S=Sequence(Prod(Z, Sequence(Z), Sequence(Z), Sequence(Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20); 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`): seq(f(n), n=0..40); # Robert Israel, Dec 19 2014 MATHEMATICA CoefficientList[Series[(-1+x)^3/(-1+4*x-3*x^2+x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 22 2012 *) LinearRecurrence[{4, -3, 1}, {1, 1, 4, 13}, 30] (* Harvey P. Dale, Oct 04 2015 *) PROG (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 (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 (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 CROSSREFS Cf. A001906, A052530, A055991, A095263. Trisection of A000930. First differences of A052544. Row sums of triangle A127893. Sequence in context: A320563 A268989 A190214 * A049222 A239249 A141364 Adjacent sequences:  A052526 A052527 A052528 * A052530 A052531 A052532 KEYWORD nonn,easy AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 STATUS approved

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Last modified October 5 13:40 EDT 2022. Contains 357258 sequences. (Running on oeis4.)