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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.
11
1, 1, 723668784231, 2827767747950, 1193097790725426305663064, 17520037013918467453246138, 7392624504986931437972335103490414473, 395235071756082109802989440265119512888, 218243704050866770455587351635302655565432102527624
OFFSET
20,3
LINKS
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.
FORMULA
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).
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).
MATHEMATICA
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_]:= 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}];
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, 10], {n, 20, 40}] (* G. C. Greubel, Nov 16 2019 *)
KEYWORD
nonn
AUTHOR
Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004
EXTENSIONS
More terms added by G. C. Greubel, Nov 16 2019
STATUS
approved