%I #47 Sep 14 2024 02:26:27
%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 numbers minus 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="https://doi.org/10.37236/665">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.
%H Murray Tannock, <a href="http://hdl.handle.net/1946/25589">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) = A000108(n) - 2.
%F a(n) = (2n)!/(n!*(n+1)!) - 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}]
%t CatalanNumber[Range[0,30]]-2 (* _Harvey P. Dale_, May 03 2019 *)
%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.
%Y Cf. A243882, A243820.
%K sign,easy,changed
%O 0,4
%A _Alexander Adamchuk_, Jul 13 2006