A092379 - OEIS

A092379

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 seven 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 #32 Nov 17 2019 02:26:23

%S 1,1,209295261,810375650,130981854694547781,1866712378783655407,

%T 380792413068640291929758918,19226936188283951521093833164,

%U 6245082121880029165837197634771465822,1084566535537396419423204907970597478243

%N 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 seven 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 G. C. Greubel, <a href="/A092379/b092379.txt">Table of n, a(n) for n = 14..60</a>

%H 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, (2003) pp. 259-264.

%F 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 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, 7).

%t M[n_, k_]:= Table[Binomial[i+j-2, i-1], {i, n}, {j, k}];

%t c[k_, n_]:= Coefficient[CharacteristicPolynomial[M[n, n], x], x, k]//Abs;

%t 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 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 Table[Q[n, 7], {n, 14, 30}] (* _Jean-François Alcover_, Sep 11 2012; modified by _G. C. Greubel_, Nov 15 2019 *)

%Y Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092378, A092380, A092381, A092382.

%K nonn

%O 14,3

%A Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004

%E More terms added and edited by _G. C. Greubel_, Nov 15 2019