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 A120304 Catalan number minus 2, or ((2n)!/(n!*(n+1)!) - 2). 10
 -1, -1, 0, 3, 12, 40, 130, 427, 1428, 4860, 16794, 58784, 208010, 742898, 2674438, 9694843, 35357668, 129644788, 477638698, 1767263188, 6564120418, 24466267018, 91482563638, 343059613648, 1289904147322, 4861946401450, 18367353072150, 69533550916002, 263747951750358 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178. Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2. [The sequence here begins 1, 1, 1, 3, 12, 40, 130, 427, 1428, 4860, ...] FORMULA a(n) = (2n)!/(n!*(n+1)!) - 2. a(n) = A000108(n) - 2. (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 MAPLE a:= n-> binomial(2*n, n)/(n+1) -2: seq(a(n), n=0..30);  # Alois P. Heinz, Jun 13 2014 MATHEMATICA Table[(2n)!/n!/(n+1)!-2, {n, 0, 30}] CatalanNumber[Range[0, 30]]-2 (* Harvey P. Dale, May 03 2019 *) PROG (MuPAD) combinat::dyckWords::count(n)-2 \$ n = 0..38; // Zerinvary Lajos, May 08 2008 (PARI) a(n) = binomial(2*n, n)/(n+1)-2; \\ Altug Alkan, Dec 17 2017 CROSSREFS Cf. A000108, A002144, A002313, A003655. Cf. A243882, A243820. Sequence in context: A308648 A247002 A027991 * A289652 A026071 A102839 Adjacent sequences:  A120301 A120302 A120303 * A120305 A120306 A120307 KEYWORD sign AUTHOR Alexander Adamchuk, Jul 13 2006 STATUS approved

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Last modified October 21 11:42 EDT 2019. Contains 328296 sequences. (Running on oeis4.)