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