A007817 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.

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

Offset

5,2

References

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.44.

R. P. Stanley, Catalan Numbers, Cambridge, 2015, p. 132.

Links

Vincenzo Librandi, Table of n, a(n) for n = 5..150

Marc Noy and Juanjo Rué, Counting polygon dissections in the projective plane, Advances Applied Math., vol.421, (2008), pp.599-619.

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

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}](* Jean-François Alcover, Nov 16 2011, after Mark van Hoeij *)

Prog

(PARI) x='x+O('x^66);
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) /* 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

Crossrefs

Sequence in context: A004408 A002409 A155655 * A285147 A327361 A293874

Adjacent sequences: A007814 A007815 A007816 * A007818 A007819 A007820

Keyword

nonn,easy,nice

Author

Victor Reiner (reiner(AT)math.umn.edu), Paul Edelman (edelman(AT)math.umn.edu)

Status

approved