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A092382 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 ten 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 A092382 #7 Nov 17 2019 02:26:09
%S A092382 1,1,723668784231,2827767747950,1193097790725426305663064,
%T A092382 17520037013918467453246138,7392624504986931437972335103490414473,
%U A092382 395235071756082109802989440265119512888,218243704050866770455587351635302655565432102527624
%N A092382 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 ten 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 A092382 G. C. Greubel, <a href="/A092382/b092382.txt">Table of n, a(n) for n = 20..70</a>
%H A092382 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 A092382 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 A092382 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 A092382 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 A092382 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, 10).
%t A092382 M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
%t A092382 c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
%t A092382 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 A092382 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 A092382 Table[Q[n, 10], {n, 20, 40}] (* _G. C. Greubel_, Nov 16 2019 *)
%Y A092382 Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092378, A092379, A092380, A092381.
%K A092382 nonn
%O A092382 20,3
%A A092382 Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004
%E A092382 More terms added by _G. C. Greubel_, Nov 16 2019