1,1

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

G. C. Greubel, Table of n, a(n) for n = 1..1000

Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.

Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, arXiv:0704.0750 [math.DG], 2007.

Branko Malesevic and I. Jovovic, The Compositions of the Differential Operations and Gateaux Directional Derivative, arXiv:0706.0249 [math.CO], 2007.

Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

a(n+3) = a(n+2) + 2*a(n+1) - a(n).

G.f.: x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3). - Ralf Stephan, Aug 19 2004

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:

LinearRecurrence[{1, 2, -1}, {5, 9, 16}, 32] (* Jean-François Alcover, Nov 22 2017 *)

(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

(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

(SageMath) 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

(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

Cf. A090989, A090991, A090992, A090993, A090994, A090995.

Cf. A000079, A007283, A020701, A020714, A129638.

Sequence in context: A020958 A020750 A020713 * A225605 A088495 A352665

Adjacent sequences: A090987 A090988 A090989 * A090991 A090992 A090993

Branko Malesevic, Feb 29 2004

More terms from Ralf Stephan, Aug 19 2004

More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007

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