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 A050489 #42 Aug 23 2025 09:19:48
%S A050489 1,11,42,155,574,2142,8052,30459,115830,442442,1696396,6525246,
%T A050489 25169452,97319900,377096040,1463921595,5692584870,22169259090,
%U A050489 86452604700,337547269290,1319388204420,5162382341220,20217646564440,79246770753150,310866899505084
%N A050489 a(n) = C(n)*(10*n + 1) where C(n) = Catalan numbers (A000108).
%D A050489 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A050489 Andrew Howroyd, <a href="/A050489/b050489.txt">Table of n, a(n) for n = 0..200</a>
%F A050489 -(n+1)*(10*n-9)*a(n) + 2*(10*n+1)*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Dec 03 2014
%F A050489 From _Stefano Spezia_, Feb 16 2020: (Start)
%F A050489 O.g.f.: 2*(1 + sqrt(1 - 4*x) + 16*x)/((1 + sqrt(1 - 4*x))^2*sqrt(1 - 4*x)).
%F A050489 E.g.f.: exp(2*x)*(I_0(2*x) + 9*I_1(2*x)), where I_n(x) is the modified Bessel function of the first kind.
%F A050489 (End)
%F A050489 G.f.: (9 - 16*x - 9*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - _Amiram Eldar_, Jul 08 2023
%F A050489 From _Peter Bala_, Aug 23 2025: (Start)
%F A050489 a(n) = binomial(2*n, n) + 9*binomial(2*n, n-1) = A000984(n) + 9*A001791(n).
%F A050489 a(n) ~ 4^n * 10/sqrt(Pi*n). (End)
%t A050489 Table[CatalanNumber[n](10n+1),{n,0,30}] (* _Harvey P. Dale_, Jul 19 2011 *)
%o A050489 (Magma) [Catalan(n)*(10*n+1):n in [0..30] ]; // _Marius A. Burtea_, Jan 05 2020
%o A050489 (PARI) a(n)=binomial(2*n,n)/(n+1)*(10*n+1) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A050489 Column k=10 of A330965.
%Y A050489 Cf. A017173, A027810, A000108.
%K A050489 easy,nonn,changed
%O A050489 0,2
%A A050489 _Barry E. Williams_, Dec 27 1999
%E A050489 Corrected and extended by _Harvey P. Dale_, Jul 19 2011