OFFSET
0,3
COMMENTS
a(n) is likewise the number of ways to place n points on an n X n grid with pairwise distinct abscissa, pairwise distinct ordinate, and mirror symmetry using one or the other of the diagonals of the grid as axis of symmetry. See also A000085 and A135401. For rotational symmetry see A001813 and A006882.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..800 (terms n = 0..99 from Manfred Scheucher)
Manfred Scheucher, Python program for enumeration
FORMULA
MAPLE
a:= proc(n) option remember; `if`(n<7, [1$2, 2, 6, 14, 46, 132][n+1],
((-25*n+149)*a(n-1)+(2*(10*n^2-7*n-106))*a(n-2)+
(45*n^2-268*n+298)*a(n-3)-(2*(10*n^2-7*n-61))*a(n-4)
-(65*n^2-367*n+522)*a(n-5)-(2*(10*n^3-67*n^2+96*n+1))*a(n-6)
-(45*n-113)*(n-4)*(n-6)*a(n-7))/(20*n-79))
end:
seq(a(n), n=0..35); # Alois P. Heinz, Jan 07 2018
MATHEMATICA
a[n_] := 2*Sum[2^k*BellB[k, 1/2]*StirlingS1[n, k], {k, 0, n}] - Sum[2^k*BellB[k]*StirlingS1[Floor[n/2], k], {k, 0, Floor[n/2]}];
Array[a, 30, 0] (* Jean-François Alcover, May 29 2019 *)
PROG
(SageMath)
def a135401(n): return sum( binomial(floor(n/2), 2*k)*binomial(2*k, k)*factorial(k)*2^(floor(n/2)-2*k) for k in range(1+floor(n/4)))
def a85(n): return sum( factorial(n) / (factorial(n-2*k) * 2^k * factorial(k)) for k in range(1+floor(n/2)))
def a297708(n): return 2*a85(n) - a135401(n)
for n in range(100): print(n, a297708(n))
CROSSREFS
KEYWORD
nonn
AUTHOR
Manfred Scheucher, Jan 03 2018
STATUS
approved