A000777 a(n) = (n+2)*Catalan(n) - 1.
1, 2, 7, 24, 83, 293, 1055, 3860, 14299, 53481, 201551, 764217, 2912167, 11143499, 42791039, 164812364, 636438059, 2463251009, 9552773999, 37112526989, 144410649239, 562724141459, 2195581527359, 8576490341249, 33537507830423, 131272552839203, 514285886020255
Offset: 0
Links
- T. D. Noe, Table of n, a(n) for n = 0..200
- Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M. On the cyclically fully commutative elements of Coxeter groups, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Table 1 type B.
- C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.
- J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1285-1332.
Programs
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Magma
[(n+2)*Catalan(n)-1: n in [0..30]]; // Vincenzo Librandi, Sep 11 2016
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Maple
[seq((binomial(2*n,n)/(n+1))*(n+2)-1,n=0..27)]; # Zerinvary Lajos, Jun 25 2006
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Mathematica
Table[(n + 2)*CatalanNumber[n] - 1, {n, 0, 20}] (* T. D. Noe, Jun 20 2012 *) -
PARI
a(n) = (n+2)*binomial(2*n,n)/(n+1) - 1; \\ Michel Marcus, Sep 11 2016
Formula
a(n) = (binomial(2*n,n)/(n+1))*(n+2) - 1. - Zerinvary Lajos, Jun 25 2006
G.f.: (1/x)*(1/2 + (6*x-1)/(2*sqrt(1-4*x))-x/(1-x)). - Vladimir Kruchinin, Aug 18 2010
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