The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A092378 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 six 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, 14061141, 54177740, 659506609478464, 9256643548177084, 155695310201316677915943, 7642657907144601059593232, 220353621720787947087602631723527 (list; graph; refs; listen; history; text; internal format)
 OFFSET 12,3 LINKS G. C. Greubel, Table of n, a(n) for n = 12..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. 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, 6). 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, 6], {n, 12, 30}] (* G. C. Greubel, Nov 15 2019 *) CROSSREFS Cf. A045912, A092372, A092373, A092374, A092375, A092376, A092377, A092379, A092380, A092381, A092382. Sequence in context: A210133 A216700 A234411 * A034634 A014497 A237141 Adjacent sequences: A092375 A092376 A092377 * A092379 A092380 A092381 KEYWORD nonn AUTHOR Saibal Mitra (smitra(AT)zonnet.nl), Mar 20 2004 STATUS approved

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

Last modified April 13 14:26 EDT 2024. Contains 371642 sequences. (Running on oeis4.)