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 A003516 M4417 #112 Aug 08 2025 09:06:57
%S A003516 1,7,36,165,715,3003,12376,50388,203490,817190,3268760,13037895,
%T A003516 51895935,206253075,818809200,3247943160,12875774670,51021117810,
%U A003516 202112640600,800472431850,3169870830126,12551759587422
%N A003516 Binomial coefficients C(2n+1, n-2).
%C A003516 a(n) is the number of royal paths (A006318) from (0,0) to (n,n) with exactly one diagonal step off the line y=x. - _David Callan_, Mar 25 2004
%C A003516 a(n) is the total number of DDUU's in all Dyck (n+2)-paths. - _David Scambler_, May 03 2013
%D A003516 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A003516 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003516 G. C. Greubel, <a href="/A003516/b003516.txt">Table of n, a(n) for n = 2..1000</a>
%H A003516 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A003516 Heidi Goodson, <a href="https://arxiv.org/abs/1901.08653">An Identity for Vertically Aligned Entries in Pascal's Triangle</a>, arXiv:1901.08653 [math.CO], 2019.
%H A003516 Milan Janjic, <a href="https://web.archive.org/web/20181001110015/https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A003516 Toufik Mansour and Mark Shattuck, <a href="https://doi.org/10.26493/2590-9770.1552.b43">Counting occurrences of subword patterns in non-crossing partitions</a>, Art Disc. Appl. Math. (2022).
%H A003516 Asamoah Nkwanta and Earl R. Barnes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Nkwanta/nkwanta2.html">Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind</a>, Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.3. - From _N. J. A. Sloane_, Sep 16 2012
%H A003516 Daniel W. Stasiuk, <a href="http://hdl.handle.net/10388/11865">An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads</a>, Master's Thesis, University of Saskatchewan-Saskatoon (2018).
%F A003516 G.f.: 32*x^2/(sqrt(1-4*x)*(sqrt(1-4*x)+1)^5). - _Marco A. Cisneros Guevara_, Jul 18 2011
%F A003516 a(n) = Sum_{k=0..n-2} binomial(n+k+2,k). - _Arkadiusz Wesolowski_, Apr 02 2012
%F A003516 D-finite with recurrence (n+3)*(n-2)*a(n) = 2*n*(2*n+1)*a(n-1). - _R. J. Mathar_, Oct 13 2012
%F A003516 G.f.: x^2*c(x)^5/sqrt(1-4*x) = ((-1 + 2*x) + (1 - 3*x + x^2) * c(x))/(x^2*sqrt(1-4*x)), with c(x) the o.g.f. of the Catalan numbers A000108. See the W. Lang link under A115139 for powers of c. - _Wolfdieter Lang_, Sep 10 2016
%F A003516 a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - _Ilya Gutkovskiy_, Sep 10 2016
%F A003516 From _Amiram Eldar_, Jan 24 2022: (Start)
%F A003516 Sum_{n>=2} 1/a(n) = 4 - 14*Pi/(9*sqrt(3)).
%F A003516 Sum_{n>=2} (-1)^n/a(n) = 228*log(phi)/(5*sqrt(5)) - 134/15, where phi is the golden ratio (A001622). (End)
%F A003516 G.f.: 2F1([7/2,3],[6],4*x). - _Karol A. Penson_, Apr 24 2024
%F A003516 a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * sqrt(x)*(x^2 - 5*x + 5)/sqrt(4 - x). - _Peter Bala_, Oct 13 2024
%e A003516 For n=4, C(2*4+1,4-2) = C(9,2) = 9*8/2 = 36, so a(4) = 36. - _Michael B. Porter_, Sep 10 2016
%t A003516 CoefficientList[ Series[ 32/(((Sqrt[1 - 4 x] + 1)^5)*Sqrt[1 - 4 x]), {x, 0, 25}], x] (* _Robert G. Wilson v_, Aug 08 2011 *)
%t A003516 Table[Binomial[2*n +1,n-2], {n,2,25}] (* _G. C. Greubel_, Jan 23 2017 *)
%o A003516 (Magma) [Binomial(2*n+1,n-2): n in [2..25]]; // _Vincenzo Librandi_, Apr 13 2011
%o A003516 (PARI) {a(n) = binomial(2*n+1, n-2)}; \\ _G. C. Greubel_, Mar 21 2019
%o A003516 (Sage) [binomial(2*n+1, n-2) for n in (2..25)] # _G. C. Greubel_, Mar 21 2019
%o A003516 (GAP) List([2..25], n-> Binomial(2*n+1, n-2)); # _G. C. Greubel_, Mar 21 2019
%Y A003516 Diagonal 6 of triangle A100257.
%Y A003516 Third unsigned column (s=2) of A113187. - _Wolfdieter Lang_, Oct 18 2012
%Y A003516 Cf. triangle A114492 - Dyck paths with k DDUU's.
%Y A003516 Cf. A001622, A115139.
%Y A003516 Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A002696 (m = 6), A030053 - A030056, A004310 - A004318.
%K A003516 nonn,easy
%O A003516 2,2
%A A003516 _N. J. A. Sloane_