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 A050478 #40 Aug 24 2025 08:44:28
%S A050478 1,9,34,125,462,1722,6468,24453,92950,354926,1360476,5231954,20177164,
%T A050478 78004500,302211720,1173076245,4561139430,17761336230,69257611500,
%U A050478 270391268070,1056823387620
%N A050478 a(n) = C(n)*(8*n+1) where C(n) = Catalan numbers (A000108).
%D A050478 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A050478 Vincenzo Librandi, <a href="/A050478/b050478.txt">Table of n, a(n) for n = 0..1000</a>
%F A050478 -(n+1)*(8*n-7)*a(n) + 2*(8*n+1)*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Dec 03 2014
%F A050478 G.f.: (7 - 12*x - 7*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - _Ilya Gutkovskiy_, Jun 13 2017
%F A050478 From _Peter Bala_, Aug 23 2025: (Start)
%F A050478 a(n) = binomial(2*n, n) + 7*binomial(2*n, n-1) = A000984(n) + 7*A001791(n).
%F A050478 a(n) ~ 2^(2*n+3)/sqrt(Pi*n). (End)
%t A050478 Table[CatalanNumber[n](8n+1),{n,0,20}] (* _Harvey P. Dale_, May 20 2012 *)
%o A050478 (Magma)[Catalan(n)*(8*n+1):n in [0..30]]; // _Vincenzo Librandi_, Jan 27 2013
%o A050478 (Magma) R<x>:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (7-12*x-7*Sqrt(1-4*x))/(2*x*Sqrt(1-4*x))))); // _Marius A. Burtea_, Jan 05 2020
%Y A050478 Column k=8 of A330965.
%Y A050478 Cf. A016993, A000108, A051945.
%K A050478 easy,nonn,changed
%O A050478 0,2
%A A050478 _Barry E. Williams_, Dec 24 1999