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A197989 Number of binary arrangements of total n 1's, without adjacent 1's on n X n array connected n-s.

%I A197989 #23 Jul 22 2025 12:54:26
%S A197989 1,4,45,886,24395,860336,36914493,1863645610,108131503623,
%T A197989 7085585223652,517329551346608,41634263983867842,3661077644199252550,
%U A197989 349191617521920855488,35902782820742394839453,3958207187579046500083794,465777357329812920074875295
%N A197989 Number of binary arrangements of total n 1's, without adjacent 1's on n X n array connected n-s.
%H A197989 Vaclav Kotesovec, <a href="/A197989/b197989.txt">Table of n, a(n) for n = 1..108</a>
%H A197989 V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, p.373-381
%F A197989 Asymptotic (V. Kotesovec, Oct 15 2011): a(n) ~ n^(2n)/n!*exp(-3/2).
%t A197989 permopak[part_,k_]:=(hist=ConstantArray[0,k];
%t A197989 Do[hist[[part[[t]]]]++,{t,1,Length[part]}];
%t A197989 (Length[part])!/Product[(hist[[t]])!,{t,1,k}]);
%t A197989 waz1n[k_,n_]:=(If[n-k+1<k,0,Binomial[n-k+1,k]]);
%t A197989 semiwaz[k_,n_]:=(psum=0;
%t A197989 Do[p=IntegerPartitions[k,{size}];
%t A197989 psum=psum+Sum[permopak[p[[i]],k]*Binomial[n,Length[p[[i]]]]*Product[waz1n[p[[i,j]],n] ,{j,1,Length[p[[i]]]}], {i,1,Length[p]}],{size,1,n}];
%t A197989 psum);
%t A197989 Table[semiwaz[n,n],{n,1,25}]
%Y A197989 Cf. A067966, A197990.
%K A197989 nonn,nice,hard
%O A197989 1,2
%A A197989 _Vaclav Kotesovec_, Oct 20 2011