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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.

%I A291717 #26 Oct 08 2017 08:56:17
%S A291717 1,4,6,9,36,8,16,120,24,168,25,300,72,714,178,36,630,144,2273,464,
%T A291717 6576,49,1176,288,5932,1476,24288,6404,64,2016,480,13536,3040,74560,
%U A291717 15680,341320,81,3240,800,27860,6940,197600,50860,1170466,314862
%N 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.
%H A291717 Andrew Howroyd, <a href="/A291717/b291717.txt">Table of n, a(n) for n = 1..1275</a>
%e A291717 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:
%e A291717   o o o X o o o
%e A291717   o o o o o o o
%e A291717   o o o o X o o
%e A291717   o X # X o o o
%e A291717   X o o o o o o
%e A291717   o o o o o o o
%e A291717   o X o o o o o
%e A291717 .
%e A291717 Triangle begins:
%e A291717    1;
%e A291717    4,    6;
%e A291717    9,   36,   8;
%e A291717   16,  120,  24,   168;
%e A291717   25,  300,  72,   714,  178;
%e A291717   36,  630, 144,  2273,  464,  6576;
%e A291717   49, 1176, 288,  5932, 1476, 24288,  6404;
%e A291717   64, 2016, 480, 13536, 3040, 74560, 15680, 341320;
%t A291717 decentralize[v_] := 2*Total[v] - Last[v];
%t A291717 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}]];
%t A291717 Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 08 2017, after _Andrew Howroyd_ *)
%o A291717 (PARI)
%o A291717 decentralize(v) = 2*vecsum(v) - v[length(v)];
%o A291717 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))))));
%o A291717 for(n=1,10, for(k=1,n, print1(T(n,k), ", ")); print); \\ _Andrew Howroyd_, Sep 16 2017
%Y A291717 Cf. A090642, A098485, A098487, A291716, A291718, A292152, A292153, A292154, A292155, A292156.
%K A291717 nonn,tabl
%O A291717 1,2
%A A291717 _Hugo Pfoertner_, Sep 08 2017
%E A291717 Terms a(37) and beyond from _Andrew Howroyd_, Sep 16 2017