|
|
A183159
|
|
The number of partial isometries (of an n-chain) of fix zero (fix of alpha = 0). Equivalently, the number of partial derangement isometries (of an n-chain).
|
|
3
|
|
|
1, 1, 4, 12, 38, 90, 220, 460, 1018, 2022, 4304, 8376, 17566, 33922, 70756, 136260, 283682, 545790, 1135576, 2184112, 4543366, 8737626, 18174764, 34951932, 72700618, 139809430, 290804320, 559239720, 1163219438
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
a(2n) = (13*4^n-12*n^2-18*n-10)/3, n>=0.
a(2n+1) = (25*4^n-12*n^2-30*n-22)/3, n>=0.
G.f.: ( 1-2*x-3*x^4+10*x^3 ) / ( (2*x-1)*(2*x+1)*(x-1)^3 ). - Joerg Arndt, Dec 30 2010
|
|
EXAMPLE
|
a(2) = 4 because there are exactly 4 partial derangement isometries (on a 2-chain) , namely: empty map; 1-->2; 2-->1; (1,2)-->(2,1). a(3) = 12 because there are exactly 12 partial isometries (on a 3-chain) namely: empty map; 1-->2; 1-->3; 2-->1; 2-->3; 3-->1; 3-->2; (1,2)-->(2,1); (1,2)-->(2,3); (2,3)-->(1,2); (2,3)-->(3,2); (1,3)-->(3,1) - the mappings are coordinate-wise.
|
|
MAPLE
|
A183159 := proc(n) nh := floor(n/2) ; if type(n, 'even') then 13*4^nh-12*nh^2-18*nh-10; else 25*4^nh-12*nh^2-30*nh-22; end if; %/3 ; end proc:
|
|
MATHEMATICA
|
LinearRecurrence[{3, 1, -11, 12, -4}, {1, 1, 4, 12, 38}, 30] (* Harvey P. Dale, Dec 06 2015 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|