A197990 - OEIS

%I #20 Mar 27 2023 22:37:02

%S 1,1,4,27,664,19375,712536,31474709,1623421808,95752130751,

%T 6356272757680,468976366239799,38071162011854412,3372179632719015287,

%U 323631920261745650114,33452466695808298399785,3705187274710433648959456,437779689881887196512539391

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

%H Andrew Howroyd, <a href="/A197990/b197990.txt">Table of n, a(n) for n = 0..200</a>

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 408.

%F a(n) = n*binomial(n^2-n-1,n-1) + n*(-1)^n, n > 1. - _Vaclav Kotesovec_, Oct 20 2011

%t permopak[part_,k_]:=(hist=ConstantArray[0,k];

%t Do[hist[[part[[t]]]]++,{t,1,Length[part]}];

%t (Length[part])!/Product[(hist[[t]])!,{t,1,k}]);

%t waz1t[k_,n_]:=(If[n-k+1<k,0,Binomial[n-k+1,k]-Binomial[n-k-1,k-2]]);

%t semiwazt[k_,n_]:=(psum=0;

%t Do[p=IntegerPartitions[k,{size}];

%t 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 Table[semiwazt[n,n],{n,1,25}]

%t Join[{1},Table[n Binomial[n^2-n-1,n-1]+n (-1)^n,{n,2,20}]] (* _Harvey P. Dale_, Nov 24 2016 *)

%o (PARI) a(n) = if(n<=1, 1, n*binomial(n^2-n-1,n-1) + n*(-1)^n) \\ _Andrew Howroyd_, Mar 27 2023

%Y Cf. A067961, A197989.

%K nonn,nice

%O 0,3

%A _Vaclav Kotesovec_, Oct 20 2011

%E a(0)=1 prepended by _Andrew Howroyd_, Mar 27 2023