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 A050477 #35 Aug 23 2025 09:19:38
%S A050477 1,8,30,110,406,1512,5676,21450,81510,311168,1192516,4585308,17681020,
%T A050477 68346800,264769560,1027653570,3995416710,15557374800,60660114900,
%U A050477 236813267460,925540979220,3621007518960,14179797364200,55575657411300,217993800897756,855702566655552
%N A050477 a(n) = C(n)*(7*n + 1) where C(n) = Catalan numbers (A000108).
%D A050477 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A050477 Andrew Howroyd, <a href="/A050477/b050477.txt">Table of n, a(n) for n = 0..200</a>
%F A050477 3*(n+1)*a(n) + (-17*n-1)*a(n-1) + 10*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Feb 13 2015
%F A050477 -(n+1)*(7*n-6)*a(n) + 2*(7*n+1)*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Feb 13 2015
%F A050477 G.f.: (3 - 5*x - 3*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - _Ilya Gutkovskiy_, Jun 13 2017
%F A050477 From _Peter Bala_, Aug 23 2025: (Start)
%F A050477 a(n) = binomial(2*n, n) + 6*binomial(2*n, n-1) = A000984(n) + 6*A001791(n).
%F A050477 a(n) ~ 4^n * 7/sqrt(Pi*n). (End)
%t A050477 Table[CatalanNumber[n](7n+1),{n,0,30}] (* _Harvey P. Dale_, Jun 01 2024 *)
%o A050477 (PARI) a(n) = (7*n+1) * binomial(2*n,n)/(n+1) \\ _Michel Marcus_, Jul 24 2013
%o A050477 (Magma) [Catalan(n)*(7*n+1):n in [0..25] ]; // _Marius A. Burtea_, Jan 05 2020
%o A050477 (Magma) R<x>:=PowerSeriesRing(Rationals(),27); (Coefficients(R!( (3-5*x-3*Sqrt(1-4*x))/(x*Sqrt(1 - 4*x))) )); // _Marius A. Burtea_, Jan 05 2020
%Y A050477 Column k=7 of A330965.
%Y A050477 Cf. A016921, A000108, A051945.
%K A050477 easy,nonn,changed
%O A050477 0,2
%A A050477 _Barry E. Williams_, Dec 24 1999
%E A050477 Terms a(21) and beyond from _Andrew Howroyd_, Jan 02 2020