OFFSET
1,1
COMMENTS
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
LINKS
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).
FORMULA
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
MAPLE
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:
MATHEMATICA
LinearRecurrence[{1, 2, -1}, {5, 9, 16}, 32] (* Jean-François Alcover, Nov 22 2017 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Branko Malesevic, Feb 29 2004
EXTENSIONS
More terms from Ralf Stephan, Aug 19 2004
More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007
STATUS
approved