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A129639
Number of meaningful differential operations of the k-th order on the space R^12.
4
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
12,1
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
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]
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 *)
KEYWORD
nonn
AUTHOR
Branko Malesevic, May 31 2007
EXTENSIONS
More terms from Joseph Myers, Dec 23 2008
STATUS
approved