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A092381 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 nine 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 A092381 #6 Nov 16 2019 20:06:36
%S A092381 1,1,47564380971,185410909790,5599434135148010392903,
%T A092381 81562945655108319592717,2647122748975437613370942794822122,
%U A092381 139318635878972598351963980703033608,6292966726927006717847495753884145618797281792
%N A092381 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 nine 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 A092381 G. C. Greubel, <a href="/A092381/b092381.txt">Table of n, a(n) for n = 18..60</a>
%H A092381 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 A092381 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 A092381 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 A092381 Even n: Q(n, m) = Sum_{r=0..(n-2*m)/4} (-1)^r * ((m+2*r)/(m+r)) * binomial(m+r, r) * C_{n/2-m- 2*r}(n).
%F A092381 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, 9).
%t A092381 M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
%t A092381 c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
%t A092381 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 A092381 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 A092381 Table[Q[n, 9], {n, 18, 40}] (* _G. C. Greubel_, Nov 16 2019 *)
%Y A092381 Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092378, A092379, A092380, A092382.
%K A092381 nonn
%O A092381 18,3
%A A092381 Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004
%E A092381 More terms added by _G. C. Greubel_, Nov 16 2019