The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.



(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A092372 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 zero 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, 3, 8, 70, 526, 13167, 280772, 20048886, 1215446794, 247358122583, 42663813089328, 24736951705389664, 12142696908022734304, 20054892679528741176540, 28022410984084414473869168 (list; graph; refs; listen; history; text; internal format)



G. C. Greubel, Table of n, a(n) for n = 1..50

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) = 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, 0).


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, 0], {n, 1, 20}] (* G. C. Greubel, Nov 15 2019 *)


Cf. A045912, A092373, A092374, A092375, A092376, A092377, A092378, A092379, A092380, A092381, A092382.

Sequence in context: A053740 A134173 A095051 * A208817 A060752 A287389

Adjacent sequences:  A092369 A092370 A092371 * A092373 A092374 A092375




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



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 27 01:40 EDT 2021. Contains 346302 sequences. (Running on oeis4.)