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A291717
Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a central symmetry.
10
1, 4, 6, 9, 36, 8, 16, 120, 24, 168, 25, 300, 72, 714, 178, 36, 630, 144, 2273, 464, 6576, 49, 1176, 288, 5932, 1476, 24288, 6404, 64, 2016, 480, 13536, 3040, 74560, 15680, 341320, 81, 3240, 800, 27860, 6940, 197600, 50860, 1170466, 314862
OFFSET
1,2
LINKS
EXAMPLE
A configuration of 6 picked points from a 7 X 7 grid with a central (point) symmetry w.r.t. point #, but no line (mirror) symmetry and thus only contributing to T(7,6)=a(27), but not to A291718(27), would be:
o o o X o o o
o o o o o o o
o o o o X o o
o X # X o o o
X o o o o o o
o o o o o o o
o X o o o o o
.
Triangle begins:
1;
4, 6;
9, 36, 8;
16, 120, 24, 168;
25, 300, 72, 714, 178;
36, 630, 144, 2273, 464, 6576;
49, 1176, 288, 5932, 1476, 24288, 6404;
64, 2016, 480, 13536, 3040, 74560, 15680, 341320;
MATHEMATICA
decentralize[v_] := 2*Total[v] - Last[v];
T[n_, k_] := decentralize[ Table[ decentralize[ Table[ If[EvenQ[k] || OddQ[a*b], Binomial[ Quotient[a*b, 2], Quotient[k, 2]], 0], {b, 1, n}]], {a, 1, n}]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
PROG
(PARI)
decentralize(v) = 2*vecsum(v) - v[length(v)];
T(n, k) = decentralize(vector(n, a, decentralize(vector(n, b, if(k%2==0||a*b%2==1, binomial(a*b\2, k\2))))));
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Sep 16 2017
KEYWORD
nonn,tabl
AUTHOR
Hugo Pfoertner, Sep 08 2017
EXTENSIONS
Terms a(37) and beyond from Andrew Howroyd, Sep 16 2017
STATUS
approved