%I #37 Aug 11 2024 22:05:53
%S 1,11,42,155,574,2142,8052,30459,115830,442442,1696396,6525246,
%T 25169452,97319900,377096040,1463921595,5692584870,22169259090,
%U 86452604700,337547269290,1319388204420,5162382341220,20217646564440,79246770753150,310866899505084
%N a(n) = C(n)*(10n+1) where C(n) = Catalan numbers (A000108).
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H Andrew Howroyd, <a href="/A050489/b050489.txt">Table of n, a(n) for n = 0..200</a>
%F -(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 From _Stefano Spezia_, Feb 16 2020: (Start)
%F O.g.f.: 2*(1 + sqrt(1 - 4*x) + 16*x)/((1 + sqrt(1 - 4*x))^2*sqrt(1 - 4*x)).
%F 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 (End)
%F G.f.: (9 - 16*x - 9*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - _Amiram Eldar_, Jul 08 2023
%t Table[CatalanNumber[n](10n+1),{n,0,30}] (* _Harvey P. Dale_, Jul 19 2011 *)
%o (Magma) [Catalan(n)*(10*n+1):n in [0..30] ]; // _Marius A. Burtea_, Jan 05 2020
%o (PARI) a(n)=binomial(2*n,n)/(n+1)*(10*n+1) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y Column k=10 of A330965.
%Y Cf. A017173, A027810, A000108.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Dec 27 1999
%E Corrected and extended by _Harvey P. Dale_, Jul 19 2011