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%I #48 Jul 28 2019 19:01:23
%S 1,6,36,196,1071,5796,31395,169884,919413,4975322,26924106,145698840,
%T 788446400,4266656226,23088902733,124944995676,676136621430,
%U 3658895818470,19800020091895,107147296401684,579824822459421,3137707025200000
%N Expansion of 1 / (1 - 6*x + 20*x^3 - 15*x^4 + x^6).
%C a(n) is the number of words of length n over an alphabet of size 6 which do not contain any strictly decreasing factor (consecutive subword) of length 3.
%C Equivalently, dimensions of homogeneous components of the universal associative envelope for a certain nonassociative triple system [Bremner].
%C This is the g.f. corresponding to row 6 of A225682.
%H R. H. Hardin and N. J. Sloane, <a href="/A200782/b200782.txt">Table of n, a(n) for n = 0..239</a> [The first 210 terms were computed by R. H. Hardin]
%H M. R. Bremner, <a href="http://arxiv.org/abs/1303.0920">Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures</a>, arXiv:1303.0920 [math.RA], 2013
%H A. Burstein and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0112281">Words restricted by 3-letter generalized multipermutation patterns</a>, Annals. Combin., 7 (2003), 1-14. See Th. 3.13.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,0,-20,15,0,-1).
%F G.f.: 1 / (1 - 6*x + 20*x^3 - 15*x^4 + x^6).
%F a(n) = 6*a(n-1) - 20*a(n-3) + 15*a(n-4) - a(n-6).
%e a(n) is also the number of words of length n over an alphabet of size 6 which do not contain any strictly increasing factor of length 3. Some solutions for n=5:
%e ..5....5....0....3....2....4....3....3....3....3....0....3....3....1....0....1
%e ..1....5....0....0....4....5....1....1....3....5....1....0....2....0....3....4
%e ..3....5....1....0....4....3....1....4....5....0....1....5....1....0....0....3
%e ..0....0....0....4....1....1....1....4....2....4....1....1....2....5....4....1
%e ..1....4....2....0....0....0....1....3....1....4....3....2....2....2....4....5
%t CoefficientList[Series[1 / (1 - 6*x + 20*x^3 - 15*x^4 + x^6), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jan 26 2015 *)
%t LinearRecurrence[{6,0,-20,15,0,-1},{1,6,36,196,1071,5796},30] (* _Harvey P. Dale_, Jul 28 2019 *)
%o (PARI) Vec(1/(1-6*x+20*x^3-15*x^4+x^6) + O(x^30)) \\ _Michel Marcus_, Jan 26 2015
%Y Column 5 of A200785.
%Y G.f. corresponds to row 6 of A225682.
%K nonn,easy
%O 0,2
%A _R. H. Hardin_, Nov 22 2011
%E Entry revised by _N. J. A. Sloane_, May 17 2013, merging this with A225381
%E Typo in name corrected by _Michel Marcus_, Jan 26 2015
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