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%I #33 Sep 08 2022 08:44:28
%S 1,2,7,24,83,293,1055,3860,14299,53481,201551,764217,2912167,11143499,
%T 42791039,164812364,636438059,2463251009,9552773999,37112526989,
%U 144410649239,562724141459,2195581527359,8576490341249,33537507830423,131272552839203,514285886020255
%N a(n) = (n+2)*Catalan(n) - 1.
%H T. D. Noe, <a href="/A000777/b000777.txt">Table of n, a(n) for n = 0..200</a>
%H Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M. <a href="https://doi.org/10.1007/s10801-011-0327-z">On the cyclically fully commutative elements of Coxeter groups</a>, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Table 1 type B.
%H C. K. Fan, <a href="http://dx.doi.org/10.1090/S0894-0347-97-00222-1">Structure of a Hecke algebra quotient</a>, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.
%H J. R. Stembridge, <a href="http://dx.doi.org/10.1090/S0002-9947-97-01805-9">Some combinatorial aspects of reduced words in finite Coxeter groups</a>, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1285-1332.
%F a(n) = (binomial(2*n,n)/(n+1))*(n+2) - 1. - _Zerinvary Lajos_, Jun 25 2006
%F G.f.: (1/x)*(1/2 + (6*x-1)/(2*sqrt(1-4*x))-x/(1-x)). - _Vladimir Kruchinin_, Aug 18 2010
%F D-finite with recurrence: (n+1)*a(n) + 4*(-3*n+1)*a(n-1) + 5*(9*n-13)*a(n-2) + 2*(-29*n+72)*a(n-3) + 12*(2*n-7)*a(n-4) = 0. - _R. J. Mathar_, Jun 11 2019
%p [seq((binomial(2*n,n)/(n+1))*(n+2)-1,n=0..27)]; # _Zerinvary Lajos_, Jun 25 2006
%t Table[(n + 2)*CatalanNumber[n] - 1, {n, 0, 20}] (* _T. D. Noe_, Jun 20 2012 *)
%o (PARI) a(n) = (n+2)*binomial(2*n,n)/(n+1) - 1; \\ _Michel Marcus_, Sep 11 2016
%o (Magma) [(n+2)*Catalan(n)-1: n in [0..30]]; // _Vincenzo Librandi_, Sep 11 2016
%Y a(n) = A038665(n-1) - 1.
%Y Cf. A000984, A000108.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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