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A092372 The O(1) loop model on the square lattice is defined as follows: At every vertex the loop turns to the left or to the right with equal probability, unless the vertex has been visited before, in which case the loop leaves the vertex via the unused edge. Every vertex is visited twice. The probability that a face of the lattice on an n X infinity cylinder is surrounded by zero loops is conjectured to be given by a(n)/A_{HT}(n)^2, where A_{HT}(n) is the number of n X n half turn symmetric alternating sign matrices.

%I A092372 #25 Nov 17 2019 13:32:32
%S A092372 1,3,8,70,526,13167,280772,20048886,1215446794,247358122583,
%T A092372 42663813089328,24736951705389664,12142696908022734304,
%U A092372 20054892679528741176540,28022410984084414473869168
%N A092372 The O(1) loop model on the square lattice is defined as follows: At every vertex the loop turns to the left or to the right with equal probability, unless the vertex has been visited before, in which case the loop leaves the vertex via the unused edge. Every vertex is visited twice. The probability that a face of the lattice on an n X infinity cylinder is surrounded by zero loops is conjectured to be given by a(n)/A_{HT}(n)^2, where A_{HT}(n) is the number of n X n half turn symmetric alternating sign matrices.
%H A092372 G. C. Greubel, <a href="/A092372/b092372.txt">Table of n, a(n) for n = 1..50</a>
%H A092372 Saibal Mitra and Bernard Nienhuis, <a href="https://dmtcs.episciences.org/3320">Osculating Random Walks on Cylinders</a>, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 259-264.
%H A092372 Saibal Mitra and Bernard Nienhuis, <a href="http://arXiv.org/abs/cond-mat/0407578">Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders</a>, arXiv:cond-mat/0407578 [cond-mat.stat-mech], 2004.
%H A092372 Saibal Mitra and Bernard Nienhuis, <a href="https://arxiv.org/abs/math-ph/0312036">Osculating Random Walks on Cylinders</a>, arXiv:math-ph/0312036, 2003.
%F A092372 Even n: Q(n, m) = C_{n/2-m}(n) + Sum_{r=1..(n-2*m)/4} (-1)^r * ((m+2*r)/(m+r)) * binomial(m+r, r) * C_{n/2-m- 2*r}(n).
%F A092372 Odd n: Q(n, m) = Sum_{r=0..(n-2*m-1)/4)} (-1)^r * binomial(m+r,r) * ( C_{(n-1)/2 -m-2*r}(n) - C_{(n-1)/2 -m-2*r-1}(n) ), where the c_{k}(n) are the absolute values of the coefficients of the characteristic polynomial of the n X n Pascal matrix P_{i, j} = binomial(i+j-2, i-1). The sequence is given by Q(n, 0).
%t A092372 M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
%t A092372 c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
%t A092372 Q[n_?EvenQ, m_]:= c[(n-2*m)/2, n] + Sum[(-1)^r*((m+2*r)/(m+r))*Binomial[m +r, r]*c[n/2 -m-2*r, n], {r, (n-2*m)/4}];
%t A092372 Q[n_?OddQ, m_]:= Sum[(-1)^r*Binomial[m+r, r]*(c[(n-1)/2 -m-2*r, n] - c[(n-1)/2 -m-2*r-1, n]), {r, 0, (n-2*m-1)/4}];
%t A092372 Table[Q[n, 0], {n, 1, 20}] (* _G. C. Greubel_, Nov 15 2019 *)
%Y A092372 Cf. A045912, A092373, A092374, A092375, A092376, A092377, A092378, A092379, A092380, A092381, A092382.
%K A092372 nonn
%O A092372 1,2
%A A092372 Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004