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%I #19 Oct 10 2017 10:44:14
%S 11,21,40,77,148,286,552,1069,2068,4010,7768,15074,29225,56736,110055,
%T 213705,414676,805314,1562977,3035514,5892257,11443768,22215753,
%U 43146726,83766396,162686691,315860810,613439352,1191054193,2313133481
%N Number of meaningful differential operations of the k-th order on the space R^11.
%C Also number of meaningful compositions of the k-th order of the differential operations and Gateaux directional derivative on the space R^10. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 20 2007
%C Also (starting 6,11,...) the number of zig-zag paths from top to bottom of a rectangle of width 12, whose color is that of the top right corner. [_Joseph Myers_, Dec 23 2008]
%H B. Malesevic, <a href="http://www.jstor.org/stable/43666958">Some combinatorial aspects of differential operation composition on the space R^n</a>, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
%H B. Malesevic and I. Jovovic, <a href="http://arXiv.org/abs/0706.0249">The Compositions of the Differential Operations and Gateaux Directional Derivative </a>, arXiv:0706.0249 [math.CO], 2007.
%H Joseph Myers, <a href="http://www.polyomino.org.uk/publications/2008/bmo1-2009-q1.pdf">BMO 2008--2009 Round 1 Problem 1---Generalisation</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-4,-6,3,1).
%F a(k+6) = a(k+5) +5*a(k+4) -4*a(k+3) -6*a(k+2) +3*a(k+1) +a(k).
%F G.f.: -x^11*(6*x^5+21*x^4-24*x^3-36*x^2+10*x+11)/(x^6+3*x^5-6*x^4-4*x^3+5*x^2+x-1). [_Colin Barker_, Jul 08 2012]
%p NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n:=11; # <- DIMENSION Fun:=(i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity:=(i,j)->piecewise(i=j,1,0); v:=matrix(1,n,1); A:=piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:
%t LinearRecurrence[{1, 5, -4, -6, 3, 1}, {11, 21, 40, 77, 148, 286}, 30] (* _Jean-François Alcover_, Oct 10 2017 *)
%Y Cf. A090989-A090995.
%Y Cf. A000079, A007283, A020701, A020714.
%K nonn,easy
%O 11,1
%A _Branko Malesevic_, May 31 2007
%E More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 20 2007
%E More terms from _Joseph Myers_, Dec 23 2008
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