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A007817
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Number of abstract simplicial 2-complexes on {1,2,3,...,n+4} which triangulate a Moebius band in such a way that all vertices lie on the boundary and are traversed in the order 1,2,3,... as one goes around the boundary.
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1
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1, 14, 113, 720, 4033, 20864, 102356, 483680, 2223482, 10009570, 44330931, 193798624, 838329841, 3595080184, 15305823256, 64766503744, 272635026526, 1142528179324, 4769415499234, 19842220567264, 82303947852506, 340491603805344, 1405318295426488, 5788074933453632, 23794580648906708, 97653338015578634, 400157876088981431
(list; graph; refs; listen; history; internal format)
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OFFSET
| 5,2
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REFERENCES
| R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.44.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 5..150
Marc Noy and Juanjo Rué Counting polygon dissections in the projective plane preprint. Appeared in Advances Applied Math., vol.421, (2008), pp.599-619.
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FORMULA
| G.f.: x^2*((2-5*x-4*x^2)+sqrt(1-4*x)*(-2+x+2*x^2))/((1-4*x)*(1-4*x+2*x^2+sqrt(1-4*x)*(1-2*x))) [from the Stanley reference, Joerg Arndt, Apr 20 2011]
a(n) = 4^(n-1)-2*(29*n^3-77*n^2+106*n-88)*binomial(2*n-5,n-1)/((n-3)*(n+1)*(n+2)). - Mark van Hoeij, Oct 30 2011
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MATHEMATICA
| a[n_] := a[n] = (4^n*(n-4)(n-3)(n*(29n-144) + 100) + 16n*(n*(n*(n*(58n-299) + 597) - 706) + 440)*a[n-1])/(8(n-1)(n+2)(n*(n*(29n-164) + 347) - 300)) ; a[5] = 1; Table[a[n], {n, 5, 31}](* From Jean-François Alcover, Nov 16 2011, after Mark van Hoeij *)
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PROG
| (Pari) x='x+O('x^66) /* that many terms */
gf=x^2*((2-5*x-4*x^2)+sqrt(1-4*x)*(-2+x+2*x^2))/((1-4*x)*(1-4*x+2*x^2+sqrt(1-4*x)*(1-2*x)));
Vec(gf) /* show terms */ /* Joerg Arndt, Apr 20 2011 */
(MAGMA) [4^(n-1)-2*(29*n^3-77*n^2+106*n-88)*Binomial(2*n-5, n-1)/((n-3)*(n+1)*(n+2)) : n in [5..30]]; // Vincenzo Librandi, Nov 17 2011
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CROSSREFS
| Sequence in context: A004408 A002409 A155655 * A044346 A044727 A199251
Adjacent sequences: A007814 A007815 A007816 * A007818 A007819 A007820
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Victor Reiner (reiner(AT)math.umn.edu), Paul Edelman (edelman(AT)math.umn.edu).
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