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 A000782 a(n) = 2*Catalan(n) - Catalan(n-1). 5
 1, 3, 8, 23, 70, 222, 726, 2431, 8294, 28730, 100776, 357238, 1277788, 4605980, 16715250, 61020495, 223931910, 825632610, 3056887680, 11360977650, 42368413620, 158498860260, 594636663660, 2236748680998, 8433988655580, 31872759742852, 120699748759856 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of Dyck (n+1)-paths that have a leading or trailing hill. - David Scambler, Aug 22 2012 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy] Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020. J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349(4) (1997), 1285-1332. FORMULA Expansion of x*(1 + x*C)*C^2, where C = (1 - (1 - 4*x)^(1/2))/(2*x) is the g.f. for the Catalan numbers, A000108. Also, expansion of (1 + x^2*C^2)*C - 1, where C = (1 - (1 - 4*x)^(1/2))/(2*x) is the g.f. for Catalan numbers, A000108. a(n) = (7*n - 5)/(n + 1) * C(n-1), where C(n) = A000108(n). - Ralf Stephan, Jan 13 2004 a(n) = leftmost column term of M^(n-1)*V, where M is a tridiagonal matrix with 1's in the super- and sub-diagonals, (1, 2, 2, 2, ...) in the main diagonal, and the rest zeros; and V is the vector [1, 2, 0, 0, 0,...]. - Gary W. Adamson, Jun 16 2011 a(n) = A000108(n+1) - A026012(n-1). - David Scambler, Aug 22 2012 MATHEMATICA CoefficientList[Series[(1+x*(1-(1-4*x)^(1/2))/(2*x)^1)*((1-(1-4*x)^(1/2))/(2*x))^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 10 2012 *) PROG (MAGMA) [2*Catalan(n)-Catalan(n-1): n in [1..30]]; // Vincenzo Librandi, Jun 10 2012 CROSSREFS Partial sums of A071735. Essentially the same as A061557. Cf. A000108, A026012. Sequence in context: A184120 A215512 A061557 * A148775 A148776 A127385 Adjacent sequences:  A000779 A000780 A000781 * A000783 A000784 A000785 KEYWORD nonn AUTHOR STATUS approved

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Last modified January 20 16:41 EST 2021. Contains 340302 sequences. (Running on oeis4.)