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A092380 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 eight 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. 11
1, 1, 3143981871, 12219117170, 26773657259138210984, 386199802888523031294, 982474651752126202075575490369, 50748123995890025746709567402256, 191795630733414647568032678703215924098176 (list; graph; refs; listen; history; text; internal format)



G. C. Greubel, Table of n, a(n) for n = 16..60

Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 259-264.

Saibal Mitra and Bernard Nienhuis, Exact conjectured expressions for correlations in the dense O(1) loop model on cylinders, arXiv:cond-mat/0407578 [cond-mat.stat-mech], 2004.

Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders, arXiv:math-ph/0312036, 2003.


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).

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, 8).


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

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

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}];

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}];

Table[Q[n, 8], {n, 16, 40}] (* G. C. Greubel, Nov 16 2019 *)


Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092378, A092389, A092381, A092382.

Sequence in context: A257893 A015397 A291600 * A096566 A217051 A198863

Adjacent sequences:  A092377 A092378 A092379 * A092381 A092382 A092383




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


More terms added by G. C. Greubel, Nov 16 2019



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Last modified August 6 19:52 EDT 2020. Contains 336256 sequences. (Running on oeis4.)