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A092373 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 one loop 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, 29, 98, 6081, 63697, 9938153, 312541502, 129127963303, 12001054360838, 13446619579882992, 3659571122336231532, 11267548349231085351832, 8927178836248655700988852, 76148331063818213217859922220 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 2..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.

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, 1).

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

CROSSREFS

Cf. A045912, A092372, A092374, A092375, A092376, A092377, A092378, A092379, A092380, A092381, A092382.

Sequence in context: A044597 A138625 A154405 * A240954 A087641 A161665

Adjacent sequences:  A092370 A092371 A092372 * A092374 A092375 A092376

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)