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Search: id:A092373
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| A092373 |
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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. |
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| 1, 1, 29, 98, 6081, 63697, 9938153, 312541502, 129127963303, 12001054360838, 13446619579882992, 3659571122336231532, 11267548349231085351832, 8927178836248655700988852, 76148331063818213217859922220
(list; graph; listen)
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OFFSET
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2,3
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REFERENCES
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Saibal Mitra and Bernard Nienhuis (2003), 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.
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LINKS
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Saibal Mitra and Bernard Nienhuis, Osculating Random Walks on Cylinders
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FORMULA
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Even n: Q(n, m)=C_{L/2-m}(n)+sum_{r=1}^{n/4-m/2}(-1)^{r}C_{n/2-m- 2r}(n)(frac{m+2r}{m+r}binom{m+r}{r}. Odd n: Q(n, m)=sum_{r=0}^{frac{(n-1)}{4}-frac{m}{2}}(-1)^{r}[C_{frac{(n-1)}{2}-m-2r}(n)-C_{frac{(n-1)}{2}-m-2r-1} (n)]binom{m+r}{r} 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}=Binom{i+j-2}{i-1}. The sequence is given by Q(n, 1)
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CROSSREFS
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Sequence in context: A044597 A138625 A141864 this_sequence A087641 A127464 A142109
Adjacent sequences: A092370 A092371 A092372 this_sequence A092374 A092375 A092376
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KEYWORD
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nonn
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AUTHOR
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Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004
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