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 A030053 #59 Oct 14 2024 10:56:32
%S A030053 1,9,55,286,1365,6188,27132,116280,490314,2042975,8436285,34597290,
%T A030053 141120525,573166440,2319959400,9364199760,37711260990,151584480450,
%U A030053 608359048206,2438362177020,9762479679106,39049918716424,156077261327400,623404249591760
%N A030053 a(n) = binomial(2n+1,n-3).
%C A030053 Number of UUUUUU's in all Dyck (n+3)-paths. - _David Scambler_, May 03 2013
%H A030053 T. D. Noe, <a href="/A030053/b030053.txt">Table of n, a(n) for n = 3..200</a>
%H A030053 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 A030053 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.
%F A030053 G.f.: x^3*128/((1-sqrt(1-4*x))^7*sqrt(1-4*x))+(-1/x^4+5/x^3-6/x^2+1/x). - _Vladimir Kruchinin_, Aug 11 2015
%F A030053 D-finite with recurrence: -(n+4)*(n-3)*a(n) +2*n*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 28 2020
%F A030053 G.f.: x^3* 2F1(4,9/2;8;4x). - _R. J. Mathar_, Jan 28 2020
%F A030053 From _Amiram Eldar_, Jan 24 2022: (Start)
%F A030053 Sum_{n>=3} 1/a(n) = 22*Pi/(9*sqrt(3)) - 33/10.
%F A030053 Sum_{n>=3} (-1)^(n+1)/a(n) = 852*log(phi)/(5*sqrt(5)) - 1073/30, where phi is the golden ratio (A001622). (End)
%F A030053 From _Peter Bala_, Oct 13 2024: (Start)
%F A030053 a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = (1/(2*Pi)) * sqrt(x)*(x^3 - 7*x^2 + 14*x - 7)/sqrt((4 - x)).
%F A030053 G.f. x^3 * B(x) * C(x)^7, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%e A030053 G.f. = x^3 + 9*x^4 + 55*x^5 + 286*x^6 + 1365*x^7 + 6188*x68 + ...
%t A030053 Table[Binomial[2*n + 1, n - 3], {n, 3, 20}] (* _T. D. Noe_, Apr 03 2014 *)
%t A030053 Rest[Rest[Rest[CoefficientList[Series[128 x^3 / ((1 - Sqrt[1 - 4 x])^7 Sqrt[1 - 4 x]) + (-1 / x^4 + 5 / x^3 - 6 / x^2 + 1 / x), {x, 0, 40}], x]]]] (* _Vincenzo Librandi_, Aug 11 2015 *)
%o A030053 (PARI) a(n) = binomial(2*n+1,n-3); \\ _Joerg Arndt_, May 08 2013
%o A030053 (Magma) [Binomial(2*n+1,n-3): n in [3..30]]; // _Vincenzo Librandi_, Aug 11 2015
%Y A030053 Diagonal 8 of triangle A100257.
%Y A030053 Cf. A001622, A113187 (unsigned fourth column).
%Y A030053 Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030054 - A030056, A004310 - A004318.
%K A030053 nonn,easy
%O A030053 3,2
%A A030053 _N. J. A. Sloane_