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A092374 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 two 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, 351, 1274, 744189, 8947743, 11416135802, 434427086992, 1338566241796974, 157000849238433534, 1228161523785291020355, 436532099633273680844304, 8925012390072153509699100030, 9502129655604190413091924623054 (list; graph; refs; listen; history; text; internal format)



G. C. Greubel, Table of n, a(n) for n = 4..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.


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, 2).


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, 2], {n, 4, 20}] (* G. C. Greubel, Nov 15 2019 *)


Cf. A045912, A092372, A092373, A092375, A092376, A092377, A092378, A092379, A092380, A092381, A092382.

Sequence in context: A292990 A281555 A267939 * A273255 A264426 A231707

Adjacent sequences:  A092371 A092372 A092373 * A092375 A092376 A092377




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



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Last modified September 27 17:05 EDT 2021. Contains 347693 sequences. (Running on oeis4.)