cp's OEIS Frontend

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

%I A129639 #17 Apr 21 2020 08:43:03
%S A129639 12,22,40,74,136,252,464,860,1584,2936,5408,10024,18464,34224,63040,
%T A129639 116848,215232,398944,734848,1362080,2508928,4650432,8566016,15877568,
%U A129639 29246208,54209408,99852800,185082496,340918784,631911168,1163969536
%N A129639 Number of meaningful differential operations of the k-th order on the space R^12.
%C A129639 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]
%H A129639 B. Malesevic, <a href="https://www.jstor.org/stable/43666958">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 A129639 Branko Malesevic, <a href="http://arxiv.org/abs/0704.0750">Some combinatorial aspects of differential operation compositions on space R^n</a>, arXiv:0704.0750 [math.DG], 2007.
%H A129639 Joseph Myers, <a href="http://www.polyomino.org.uk/publications/2008/bmo1-2009-q1.pdf">BMO 2008--2009 Round 1 Problem 1---Generalisation</a>
%F A129639 f(k+6) = 6*f(k+4)-10*f(k+2)+4*f(k).
%F A129639 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]
%p A129639 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:
%t A129639 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]];
%t A129639 f /@ Range[12, 42] (* _Jean-François Alcover_, Apr 21 2020 *)
%Y A129639 Cf. A090989, A090990, A090991, A090992, A090993, A090994, A090995.
%K A129639 nonn
%O A129639 12,1
%A A129639 _Branko Malesevic_, May 31 2007
%E A129639 More terms from _Joseph Myers_, Dec 23 2008