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%I #34 Mar 27 2024 08:58:20
%S 5,9,16,29,52,94,169,305,549,990,1783,3214,5790,10435,18801,33881,
%T 61048,110009,198224,357194,643633,1159797,2089869,3765830,6785771,
%U 12227562,22033274,39702627,71541613,128913593,232294192,418579765
%N Number of meaningful differential operations of the n-th order on the space R^5.
%C Also number of meaningful compositions of the n-th order of the differential operations and Gateaux directional derivative on the space R^4. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007
%H G. C. Greubel, <a href="/A090990/b090990.txt">Table of n, a(n) for n = 1..1000</a>
%H Branko Malesevic, <a href="http://pefmath2.etf.rs/files/118/869.pdf">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 Branko Malesevic, <a href="https://arxiv.org/abs/0704.0750">Some combinatorial aspects of differential operation composition on the space R^n</a>, arXiv:0704.0750 [math.DG], 2007.
%H Branko Malesevic and I. Jovovic, <a href="https://arxiv.org/abs/0706.0249">The Compositions of the Differential Operations and Gateaux Directional Derivative</a>, arXiv:0706.0249 [math.CO], 2007.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1).
%F a(n+3) = a(n+2) + 2*a(n+1) - a(n).
%F G.f.: x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3). - _Ralf Stephan_, Aug 19 2004
%p NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 5; # <- 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, 2, -1}, {5, 9, 16}, 32] (* _Jean-François Alcover_, Nov 22 2017 *)
%o (PARI) my(x='x+O('x^40)); Vec(x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3)) \\ _G. C. Greubel_, Feb 02 2019
%o (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3) )); // _G. C. Greubel_, Feb 02 2019
%o (Sage) a=(x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3)).series(x, 40).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Feb 02 2019
%o (GAP) a:=[5,9,16];; for n in [4..30] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]; od; a; # _G. C. Greubel_, Feb 02 2019
%Y Cf. A090989, A090991, A090992, A090993, A090994, A090995.
%Y Cf. A000079, A007283, A020701, A020714, A129638.
%K nonn
%O 1,1
%A _Branko Malesevic_, Feb 29 2004
%E More terms from _Ralf Stephan_, Aug 19 2004
%E More terms from _Branko Malesevic_ and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007
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