This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000782 #60 Apr 05 2025 10:12:18
%S A000782 1,3,8,23,70,222,726,2431,8294,28730,100776,357238,1277788,4605980,
%T A000782 16715250,61020495,223931910,825632610,3056887680,11360977650,
%U A000782 42368413620,158498860260,594636663660,2236748680998,8433988655580,31872759742852,120699748759856
%N A000782 a(n) = 2*Catalan(n) - Catalan(n-1).
%C A000782 Number of Dyck (n+1)-paths that have a leading or trailing hill. - _David Scambler_, Aug 22 2012
%C A000782 a(n) is the number of parking functions of size n avoiding the patterns 132, 213, 312, and 321. - _Lara Pudwell_, Apr 10 2023
%C A000782 Number of Dyck (n+1)-paths that have exactly one return to the x-axis and/or a peak in the center of the path. - _Roger Ford_, May 15 2024
%H A000782 Vincenzo Librandi, <a href="/A000782/b000782.txt">Table of n, a(n) for n = 1..1000</a>
%H A000782 Ayomikun Adeniran and Lara Pudwell, <a href="https://doi.org/10.54550/ECA2023V3S3R17">Pattern avoidance in parking functions</a>, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
%H A000782 Ling Gao, <a href="http://hdl.handle.net/20.500.12680/h989rb533">Graph assembly for spider and tadpole graphs</a>, Master's Thesis, Cal. State Poly. Univ. (2023).
%H A000782 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A000782 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A000782 Anna Rodriguez Rasmussen, <a href="https://arxiv.org/abs/2504.01706">Exact Borel subalgebras of quasi-hereditary monomial algebras</a>, arXiv:2504.01706 [math.RT], 2025. See p. 38.
%H A000782 John R. Stembridge, <a href="http://dx.doi.org/10.1090/S0002-9947-97-01805-9">Some combinatorial aspects of reduced words in finite Coxeter groups</a>, Trans. Amer. Math. Soc. 349(4) (1997), 1285-1332.
%F A000782 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.
%F A000782 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.
%F A000782 a(n) = (7*n - 5)/(n + 1) * C(n-1), where C(n) = A000108(n). - _Ralf Stephan_, Jan 13 2004
%F A000782 a(n) = leftmost column term of M^(n-1)*V, where M is a tridiagonal matrix with 1's in the super- and subdiagonals, (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
%F A000782 a(n) = A000108(n+1) - A026012(n-1). - _David Scambler_, Aug 22 2012
%t A000782 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 *)
%o A000782 (Magma) [2*Catalan(n)-Catalan(n-1): n in [1..30]]; // _Vincenzo Librandi_, Jun 10 2012
%Y A000782 Partial sums of A071735.
%Y A000782 Essentially the same as A061557.
%Y A000782 Cf. A000108, A026012.
%K A000782 nonn
%O A000782 1,2
%A A000782 _N. J. A. Sloane_