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.

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

Author

Hugo Pfoertner, Sep 08 2017

Keywords

• nonn
• tabl

Examples

			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;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Cf. A090642, A098485, A098487, A291716, A291718, A292152, A292153, A292154, A292155, A292156.

Programs

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 *)

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

Extensions

Terms a(37) and beyond from Andrew Howroyd, Sep 16 2017