A092381 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 nine 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.

1, 1, 47564380971, 185410909790, 5599434135148010392903, 81562945655108319592717, 2647122748975437613370942794822122, 139318635878972598351963980703033608, 6292966726927006717847495753884145618797281792

Offset

18,3

Links

G. C. Greubel, Table of n, a(n) for n = 18..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.

Formula

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, 9).

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, 9], {n, 18, 40}] (* G. C. Greubel, Nov 16 2019 *)

Crossrefs

Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092378, A092379, A092380, A092382.

Sequence in context: A179227 A003940 A003933 * A195283 A344750 A344862

Adjacent sequences: A092378 A092379 A092380 * A092382 A092383 A092384

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