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.

1, 1, 209295261, 810375650, 130981854694547781, 1866712378783655407, 380792413068640291929758918, 19226936188283951521093833164, 6245082121880029165837197634771465822, 1084566535537396419423204907970597478243

Offset

14,3

Links

G. C. Greubel, Table of n, a(n) for n = 14..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, (2003) pp. 259-264.

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, 7).

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_]:= 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, 7], {n, 14, 30}] (* Jean-François Alcover, Sep 11 2012; modified by G. C. Greubel, Nov 15 2019 *)

Crossrefs

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

Sequence in context: A288088 A233493 A187440 * A233614 A105294 A288079

Adjacent sequences: A092376 A092377 A092378 * A092380 A092381 A092382

Keyword

nonn

Author

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

Extensions

More terms added and edited by G. C. Greubel, Nov 15 2019

Status

approved