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A197990 Number of binary arrangements of total n 1's, without adjacent 1's on n X n torus connected n-s.

%I A197990 #20 Mar 27 2023 22:37:02
%S A197990 1,1,4,27,664,19375,712536,31474709,1623421808,95752130751,
%T A197990 6356272757680,468976366239799,38071162011854412,3372179632719015287,
%U A197990 323631920261745650114,33452466695808298399785,3705187274710433648959456,437779689881887196512539391
%N A197990 Number of binary arrangements of total n 1's, without adjacent 1's on n X n torus connected n-s.
%H A197990 Andrew Howroyd, <a href="/A197990/b197990.txt">Table of n, a(n) for n = 0..200</a>
%H A197990 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 408.
%F A197990 a(n) = n*binomial(n^2-n-1,n-1) + n*(-1)^n, n > 1. - _Vaclav Kotesovec_, Oct 20 2011
%t A197990 permopak[part_,k_]:=(hist=ConstantArray[0,k];
%t A197990 Do[hist[[part[[t]]]]++,{t,1,Length[part]}];
%t A197990 (Length[part])!/Product[(hist[[t]])!,{t,1,k}]);
%t A197990 waz1t[k_,n_]:=(If[n-k+1<k,0,Binomial[n-k+1,k]-Binomial[n-k-1,k-2]]);
%t A197990 semiwazt[k_,n_]:=(psum=0;
%t A197990 Do[p=IntegerPartitions[k,{size}];
%t A197990 psum=psum+Sum[permopak[p[[i]],k]*Binomial[n,Length[p[[i]]]]*Product[waz1t[p[[i,j]],n], {j,1,Length[p[[i]]]}], {i,1,Length[p]}], {size,1,n}]; psum);
%t A197990 Table[semiwazt[n,n],{n,1,25}]
%t A197990 Join[{1},Table[n Binomial[n^2-n-1,n-1]+n (-1)^n,{n,2,20}]] (* _Harvey P. Dale_, Nov 24 2016 *)
%o A197990 (PARI) a(n) = if(n<=1, 1, n*binomial(n^2-n-1,n-1) + n*(-1)^n) \\ _Andrew Howroyd_, Mar 27 2023
%Y A197990 Cf. A067961, A197989.
%K A197990 nonn,nice
%O A197990 0,3
%A A197990 _Vaclav Kotesovec_, Oct 20 2011
%E A197990 a(0)=1 prepended by _Andrew Howroyd_, Mar 27 2023