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%I #36 Sep 08 2022 08:44:35
%S 1,14,113,720,4033,20864,102356,483680,2223482,10009570,44330931,
%T 193798624,838329841,3595080184,15305823256,64766503744,272635026526,
%U 1142528179324,4769415499234,19842220567264,82303947852506,340491603805344,1405318295426488,5788074933453632,23794580648906708,97653338015578634,400157876088981431
%N 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.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.44.
%D R. P. Stanley, Catalan Numbers, Cambridge, 2015, p. 132.
%H Vincenzo Librandi, <a href="/A007817/b007817.txt">Table of n, a(n) for n = 5..150</a>
%H Marc Noy and Juanjo Rué, <a href="https://doi.org/10.1016/j.aam.2008.03.003">Counting polygon dissections in the projective plane</a>, Advances Applied Math., vol.421, (2008), pp.599-619.
%F 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]
%F 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
%t 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_ *)
%o (PARI) x='x+O('x^66);
%o 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)));
%o Vec(gf) /* _Joerg Arndt_, Apr 20 2011 */
%o (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
%K nonn,easy,nice
%O 5,2
%A Victor Reiner (reiner(AT)math.umn.edu), Paul Edelman (edelman(AT)math.umn.edu)
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