A129639 Number of meaningful differential operations of the k-th order on the space R^12.

12, 22, 40, 74, 136, 252, 464, 860, 1584, 2936, 5408, 10024, 18464, 34224, 63040, 116848, 215232, 398944, 734848, 1362080, 2508928, 4650432, 8566016, 15877568, 29246208, 54209408, 99852800, 185082496, 340918784, 631911168, 1163969536

Offset: 12

Author

Branko Malesevic, May 31 2007

Keywords

• nonn

Comments

Also (starting 7,12,...) the number of zig-zag paths from top to bottom of a rectangle of width 7. [Joseph Myers, Dec 23 2008]

Links

• 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

Crossrefs

Cf. A090989, A090990, A090991, A090992, A090993, A090994, A090995.

Programs

Maple

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n:=12; # <- 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

f[k_] := f[k] = If[k <= 17, {12, 22, 40, 74, 136, 252}[[k-11]], 6 f[k-2] - 10 f[k-4] + 4 f[k-6]];
f /@ Range[12, 42] (* Jean-François Alcover, Apr 21 2020 *)

Formula

f(k+6) = 6*f(k+4)-10*f(k+2)+4*f(k).

Empirical G.f.: 2*x^12*(6+11*x-4*x^2-7*x^3)/(1-4*x^2+2*x^4). [Colin Barker, May 07 2012]

Extensions

More terms from Joseph Myers, Dec 23 2008