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%I #25 Sep 08 2022 08:44:59
%S 1,6,22,80,294,1092,4092,15444,58630,223652,856596,3292016,12688732,
%T 49031400,189885240,736808220,2863971270,11149451940,43465121700,
%U 169657266240,662976162420,2593424304120,10154564564040,39794915183400,156078401826204,612605246582952
%N a(n) = C(n)*(5n+1) where C(n) = Catalan numbers (A000108).
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H Andrew Howroyd, <a href="/A051945/b051945.txt">Table of n, a(n) for n = 0..200</a>
%F (n+1)*(5n-4)*a(n) - 2*(5n+1)(2n-1)*a(n-1) = 0. - _R. J. Mathar_, Jul 09 2012
%F G.f.: (2 - 3*x - 2*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - _Ilya Gutkovskiy_, Jun 13 2017
%t Table[CatalanNumber[n](5n+1),{n,0,30}] (* _Harvey P. Dale_, Jul 27 2020 *)
%o (PARI) a(n) = (5*n+1)*binomial(2*n, n)/(n+1) \\ _Michel Marcus_, Jul 12 2013
%o (Magma) [Catalan(n)*(5*n+1):n in [0..27] ]; // _Marius A. Burtea_, Jan 05 2020
%o (Magma) R<x>:=PowerSeriesRing(Rationals(),29); (Coefficients(R!((2-3*x-2*Sqrt(1-4*x))/(x*Sqrt(1-4*x))))); // _Marius A. Burtea_, Jan 05 2020
%Y Column k=5 of A330965.
%Y Cf. A016813, A000108, A051924.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Dec 20 1999
%E Terms a(21) and beyond from _Andrew Howroyd_, Jan 04 2020
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