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A092374 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 two 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 A092374 #8 Nov 16 2019 13:50:34
%S A092374 1,1,351,1274,744189,8947743,11416135802,434427086992,
%T A092374 1338566241796974,157000849238433534,1228161523785291020355,
%U A092374 436532099633273680844304,8925012390072153509699100030,9502129655604190413091924623054
%N A092374 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 two 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 A092374 G. C. Greubel, <a href="/A092374/b092374.txt">Table of n, a(n) for n = 4..60</a>
%H A092374 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.
%F A092374 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 A092374 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, 2).
%t A092374 M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
%t A092374 c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
%t A092374 Q[n_?EvenQ, m_]:= Sum[(-1)^r*((m+2*r)/(m+r))*Binomial[m +r, r]*c[n/2 -m-2*r, n], {r, 0, (n-2*m)/4}];
%t A092374 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 A092374 Table[Q[n, 2], {n, 4, 20}] (* _G. C. Greubel_, Nov 15 2019 *)
%Y A092374 Cf. A045912, A092372, A092373, A092375, A092376, A092377, A092378, A092379, A092380, A092381, A092382.
%K A092374 nonn
%O A092374 4,3
%A A092374 Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004