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A092373 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 one loop 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 A092373 #9 Nov 16 2019 13:50:26
%S A092373 1,1,29,98,6081,63697,9938153,312541502,129127963303,12001054360838,
%T A092373 13446619579882992,3659571122336231532,11267548349231085351832,
%U A092373 8927178836248655700988852,76148331063818213217859922220
%N A092373 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 one loop 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 A092373 G. C. Greubel, <a href="/A092373/b092373.txt">Table of n, a(n) for n = 2..50</a>
%H A092373 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 A092373 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 A092373 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, 1).
%t A092373 M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];
%t A092373 c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;
%t A092373 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 A092373 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 A092373 Table[Q[n, 1], {n, 2, 20}] (* _G. C. Greubel_, Nov 15 2019 *)
%Y A092373 Cf. A045912, A092372, A092374, A092375, A092376, A092377, A092378, A092379, A092380, A092381, A092382.
%K A092373 nonn
%O A092373 2,3
%A A092373 Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004