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A019318 Number of inequivalent ways of choosing n squares from an n X n board, considering rotations and reflections to be the same.

%I A019318 #35 Nov 02 2021 22:21:15
%S A019318 1,2,16,252,6814,244344,10746377,553319048,32611596056,2163792255680,
%T A019318 159593799888052,12952412056879996,1147044793316531040,
%U A019318 110066314584030859544,11375695977099383509351,1259843950257390597789296,148842380543159458506703546,18685311541775061906510072648,2483858381692984848273972297368,348545122958862200122401771463328
%N A019318 Number of inequivalent ways of choosing n squares from an n X n board, considering rotations and reflections to be the same.
%C A019318 Number of n X n binary matrices with n ones under action of dihedral group of the square D_4.
%H A019318 Mathieu Gouttenoire, <a href="/A019318/b019318.txt">Table of n, a(n) for n = 1..300</a>
%H A019318 Mario Velucchi, <a href="https://web.archive.org/web/20060301112158/http://www.cli.di.unipi.it/~velucchi/diff.txt">Different Dispositions in the ChessBoard</a>.
%H A019318 Mario Velucchi, <a href="https://web.archive.org/web/20001002133944/http://anduin.eldar.org/~problemi/diff.zip">Different Dispositions in the ChessBoard</a>.
%F A019318 See Velucchi link or the PARI program. Note that the polynomial whose coefficient of a^k is divided by 8 differs based upon whether the term's index is even or odd.
%F A019318 Let A(n) = C(n^2, n); B(n) = C((n^2-(n mod 2))/2, n/2); C(n) = C((n^2-(n mod 2))/4, n/4); D(n) = Sum(p = 0 to [n/2], C((n^2-n)/2, p)*C(n, n-2p)). Then a(n) = (A(n) + 3B(n) + 2C(n) + 2D(n))/8 if n == 0 (mod 4), (A(n) + B(n) + 2C(n) + 4D(n))/8 if n == 1 (mod 4), (A(n) + 3B(n) + 2D(n))/8 if n == 2 (mod 4), (A(n) + B(n) + 4D(n))/8 if n == 3 (mod 4). - _David W. Wilson_, May 29 2003
%e A019318 For n=3 the 16 solutions are
%e A019318 111 110 110 110 110 110 110 101 101 101 100 100 100 010 010 010
%e A019318 000 100 010 001 000 000 000 010 000 000 011 010 001 110 101 010
%e A019318 000 000 000 000 100 010 001 000 100 010 000 001 010 000 000 010
%t A019318 p[a_, b_, n_] := If[EvenQ[n], (a+b)^(n^2) + 2*(a+b)^n*(a^2 + b^2)^((n^2 - n)/2) + 3*(a^2 + b^2)^(n^2/2) + 2*(a^4 + b^4)^(n^2/4), (a+b)^(n^2) + 2*(a+b)*(a^4 + b^4)^((n^2-1)/4) + (a+b)*(a^2 + b^2)^((n^2-1)/2) + 4*(a+b)^n*(a^2 + b^2)^((n^2-n)/2)]; Table[Coefficient[p[a, 1, k], a, k]/8, {k, 1, 20}] (* _Jean-François Alcover_, Nov 12 2013, translated from Pari *)
%o A019318 (PARI) {p(a,b,N) = if(N%2==0, (a+b)^(N^2) + 2*(a+b)^N*(a^2+b^2)^((N^2-N)/2) + 3*(a^2+b^2)^(N^2/2) + 2*(a^4+b^4)^(N^2/4), (a+b)^(N^2) + 2*(a+b)*(a^4+b^4)^((N^2-1)/4) + (a+b)*(a^2+b^2)^((N^2-1)/2) + 4*(a+b)^N*(a^2+b^2)^((N^2-N)/2))} for(k=1,20,print1(polcoeff(p(a,1,k),k)/8,","))
%Y A019318 Cf. A054252 and A014409.
%K A019318 nonn,nice
%O A019318 1,2
%A A019318 Mario Velucchi (mathchess(AT)velucchi.it)
%E A019318 More terms from _Rick L. Shepherd_ and _David W. Wilson_, May 28 2003