%I A122367
%S A122367 1,2,5,13,34,89,233,610,1597,4181,10946,28657,75025,196418,514229,
%T A122367 1346269,3524578,9227465,24157817,63245986,165580141,433494437,
%U A122367 1134903170,2971215073,7778742049,20365011074,53316291173,139583862445
%N A122367 Dimension of 3-variable non-commutative harmonics (twisted derivative). 
               The dimension of the space of non-commutative polynomials in 3 variables 
               which are killed by all symmetric differential operators (where for 
               a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j).
%C A122367 Essentially identical to A001519
%C A122367 Comments from Matthew Lehman (matt.comicopia(AT)gmail.com), Jun 14 2008: 
               Number of monotonic rhythms using n time intervals of equal duration 
               (starting with n=0).
%C A122367 Representationally, let O be an interval which is "off" (rest),
%C A122367 X an interval which is "on" (beep),
%C A122367 X X two consecutive "on" intervals (beep, beep),
%C A122367 X O X (beep, rest, beep) and
%C A122367 X-X two connected consecutive "on" intervals (beeeep).
%C A122367 For f(3)=13:
%C A122367 O O O, O O X, O X O, O X X, O X-X, X O O, X O X,
%C A122367 X X O, X-X O, X X X, X X-X, X-X X, X-X-X
%C A122367 Contribution from Matthew Lehman (matt.comicopia(AT)gmail.com), Nov 22 
               2008: (Start)
%C A122367 Equivalent to the number of one-dimensional graphs of n nodes,
%C A122367 subject to the condition that a node is either 'on' or 'off'
%C A122367 and that any two neighboring 'on' nodes can be connected. (End)
%C A122367 Sum_{n>=0} atan(1/a(n)) = Pi/2 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Feb 27 2009]
%D A122367 N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants 
               of the Symmetric Group in Noncommuting Variables, to appear Canad. 
               J. Math., arXiv:math.CO/0502082
%D A122367 C. Chevalley, Invariants of finite groups generated by reflections, Amer. 
               J. Math. 77 (1955), 778-782.
%D A122367 M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. 
               J. 2 (1936), 626-637.
%H A122367 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A122367 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
               fibpi.html">Pi and the Fibonacci numbers</a> [From Jaume Oliver Lafont 
               (joliverlafont(AT)gmail.com), Feb 27 2009]
%F A122367 G.f.: (1-q)/(1-3*q+q^2). More generally, sum( n!/(n-d)!*q^d/prod((1-r*q),
               r=1..d), d=0..n)/sum( q^d/prod((1-r*q),r=1..d), d=0..n) where n=3 
               a(n) = 3*a(n-1)-a(n-2) with a(0) = 1, a(1) = 2
%F A122367 a(n)=Fibonacci(2n+1)=A000045(2n+1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Feb 11 2009]
%e A122367 a(1) = 2 because x1-x2, x1-x3 are both of degree 1 and are killed by 
               the differential operator d_x1+d_x2+d_x3
%e A122367 a(2) = 5 because x1 x2 - x3 x2, x1 x3 - x2 x3, x2 x1 - x3 x1, x1 x1 - 
               x2 x1 - x2 x2 + x1 x2, x1 x1 - x3 x1 - x3 x3 + x3 x1 are all lin. 
               ind. and are killed by d_x1+d_x2+d_x3, d_x1 d_x1 + d_x2 d_x2 + d_x3 
               d_x3 and sum( d_xi d_xj, i,j = 1..3)
%p A122367 a:=n->if n=0 then 1; elif n=1 then 2 else 3*a(n-1)-a(n-2); fi;
%p A122367 a:=n->sum(binomial(n+k,2*k), k=0..n): seq(a(n), n=0..27); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007
%p A122367 with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, 
               Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, 
               Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length,
               Z), card>=0), Z, end_blockRL):Q:=subs([a=Union(ZL3), b=ZL1], ZL0), 
               begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, 
               end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, 
               mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, 
               {Q}, unlabelled], size=n), n=1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 08 2008
%Y A122367 Cf. A001519, A055105, A055107, A087903, A074664, A008277, A112340, A122368, 
               A122369, A122370, A122371, A122372.
%Y A122367 Sequence in context: A027933 A141448 A011783 this_sequence A001519 A048575 
               A099496
%Y A122367 Adjacent sequences: A122364 A122365 A122366 this_sequence A122368 A122369 
               A122370
%K A122367 nonn
%O A122367 0,2
%A A122367 Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 30 2006