A090993 Number of meaningful differential operations of the n-th order on the space R^8.

8, 14, 24, 42, 72, 126, 216, 378, 648, 1134, 1944, 3402, 5832, 10206, 17496, 30618, 52488, 91854, 157464, 275562, 472392, 826686, 1417176, 2480058, 4251528, 7440174, 12754584, 22320522, 38263752, 66961566, 114791256, 200884698

Offset

1,1

Comments

Also (starting 5,8,...) the number of zig-zag paths from top to bottom of a rectangle of width 5. - Joseph Myers, Dec 23 2008

Number of walks of length n on the path graph P_5. - Andrew Howroyd, Apr 17 2017

Links

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

B. 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 compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation

Index entries for linear recurrences with constant coefficients, signature (0,3).

Formula

a(n+4) = 4*a(n+2) - 3*a(n).

From Colin Barker, May 03 2012: (Start)

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

G.f.: 2*x*(4+7*x)/(1-3*x^2). (End)

a(n) = (11+3*(-1)^n) * 3^floor((n-1)/2). - Ralf Stephan, Jul 19 2013

Maple

NUM := proc(k :: integer) local i, j, n, Fun, Identity, v, A; n := 8; # <- 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[{0, 3}, {8, 14}, 32] (* Jean-François Alcover, Jul 01 2018 *)

Prog

(PARI) my(x='x+O('x^40)); Vec(2*x*(4+7*x)/(1-3*x^2)) \\ G. C. Greubel, Feb 02 2019
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(  2*x*(4+7*x)/(1-3*x^2) )); // G. C. Greubel, Feb 02 2019
(Sage) a=(2*x*(4+7*x)/(1-3*x^2)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 02 2019
(GAP) a:=[8, 14];; for n in [3..40] do a[n]:=3*a[n-2]; od; a; # G. C. Greubel, Feb 02 2019

Crossrefs

Cf. A090989-A090995.

Column 5 of A220062.

Sequence in context: A155156 A275898 A248427 * A211525 A241161 A014002

Adjacent sequences: A090990 A090991 A090992 * A090994 A090995 A090996

Keyword

nonn,easy

Author

Branko Malesevic, Feb 29 2004

Extensions

More terms from Joseph Myers, Dec 23 2008

Status

approved