Search: id:A052529
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%I A052529
%S A052529 1,1,4,13,41,129,406,1278,4023,12664,39865,125491,395033,1243524,
%T A052529 3914488,12322413,38789712,122106097,384377665,1209982081,3808901426,
%U A052529 11990037126,37743426307,118812495276,374009739309,1177344897715
%N A052529 A simple regular expression.
%C A052529 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) 
               ]]].
%C A052529 a(n+1) = number of unique matrix products in (A+B+C+D)^n where commutator 
               [A,B]=[A,D]=[B,D]=0 but D does not commute with A, B, or C. - Paul 
               D. Hanna and Max Alekseyev (maxale(AT)gmail.com), Feb 01 2006
%C A052529 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). [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
%D A052529 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 80.
%H A052529 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=459">
               Encyclopedia of Combinatorial Structures 459</a>
%F A052529 G.f.: (-1+x)^3/(-1+4*x-3*x^2+x^3)
%F A052529 Recurrence: a(n)=4*a(n-1)-3*a(n-2)+a(n-3) for n>=4.
%F A052529 Sum(-1/31*(5*_alpha+3*_alpha^2-6)*_alpha^(-1-n), _alpha=RootOf(-1+4*_Z-3*_Z^2+_Z^3))
%F A052529 For n>0, a(n)=sum(k=0, n-1, sum(i=0, k, sum(j=0, i, a(j)))). - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Jan 26 2003
%F A052529 a(n) = Sum_{k=0..n} binomial(n+2*k-1, n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Mar 23 2003
%p A052529 spec := [S,{S=Sequence(Prod(Z,Sequence(Z),Sequence(Z),Sequence(Z)))},
               unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%Y A052529 Cf. A001906, A055991.
%Y A052529 Trisection of A000930. First differences of A052544.
%Y A052529 Sequence in context: A097112 A077284 A070428 this_sequence A049222 A001453 
               A141364
%Y A052529 Adjacent sequences: A052526 A052527 A052528 this_sequence A052530 A052531 
               A052532
%K A052529 easy,nonn
%O A052529 0,3
%A A052529 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052529 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 08 2000

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