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A120304 Catalan number minus 2, or ((2n)!/(n!*(n+1)!) - 2). 10

%I

%S -1,-1,0,3,12,40,130,427,1428,4860,16794,58784,208010,742898,2674438,

%T 9694843,35357668,129644788,477638698,1767263188,6564120418,

%U 24466267018,91482563638,343059613648,1289904147322,4861946401450,18367353072150,69533550916002,263747951750358

%N Catalan number minus 2, or ((2n)!/(n!*(n+1)!) - 2).

%C Prime p divides a(p). Prime p divides a(p+1) for p > 2. Prime p divides a(p-1)/2) for p = 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, ... = A002144(n) except 5. Pythagorean primes: primes of form 4n+1. Also A002313(n) except 2, 5. Primes congruent to 1 or 2 modulo 4; or, primes of form x^2+y^2; or, -1 is a square mod p. p^2 divides a(p^2) and a(p^2+1) for all prime p.

%C For n >= 2, number of Dyck paths of semilength n such that all four consecutive step patterns of length 2 occur at least once; a(3)=3: UDUUDD, UUDDUD, UUDUDD. For each n two paths do not satisfy the condition: U^{n}D^{n} and (UD)^n. - _Alois P. Heinz_, Jun 13 2014

%H Vincenzo Librandi, <a href="/A120304/b120304.txt">Table of n, a(n) for n = 0..1000</a>

%H J.-L. Baril, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.

%H Murray Tannock, <a href="https://skemman.is/bitstream/1946/25589/1/msc-tannock-2016.pdf">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2. [The sequence here begins 1, 1, 1, 3, 12, 40, 130, 427, 1428, 4860, ...]

%F a(n) = (2n)!/(n!*(n+1)!) - 2. a(n) = A000108(n) - 2.

%F (n+1)*a(n) + 2*(-3*n+1)*a(n-1) + (9*n-13)*a(n-2) + 2*(-2*n+5)*a(n-3) = 0. - _R. J. Mathar_, May 30 2014

%p a:= n-> binomial(2*n, n)/(n+1) -2:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jun 13 2014

%t Table[(2n)!/n!/(n+1)!-2,{n,0,30}]

%o (MuPAD) combinat::dyckWords::count(n)-2 $ n = 0..38; // _Zerinvary Lajos_, May 08 2008

%o (PARI) a(n) = binomial(2*n, n)/(n+1)-2; \\ _Altug Alkan_, Dec 17 2017

%Y Cf. A000108, A002144, A002313, A003655.

%Y Cf. A243882, A243820.

%K sign

%O 0,4

%A _Alexander Adamchuk_, Jul 13 2006

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Last modified October 22 21:09 EDT 2018. Contains 316505 sequences. (Running on oeis4.)