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 A051945 #36 Aug 29 2025 08:43:11
%S A051945 1,6,22,80,294,1092,4092,15444,58630,223652,856596,3292016,12688732,
%T A051945 49031400,189885240,736808220,2863971270,11149451940,43465121700,
%U A051945 169657266240,662976162420,2593424304120,10154564564040,39794915183400,156078401826204,612605246582952
%N A051945 a(n) = C(n)*(5*n+1) where C(n) = Catalan numbers (A000108).
%D A051945 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A051945 Andrew Howroyd, <a href="/A051945/b051945.txt">Table of n, a(n) for n = 0..200</a>
%F A051945 (n+1)*(5n-4)*a(n) - 2*(5n+1)(2n-1)*a(n-1) = 0. - _R. J. Mathar_, Jul 09 2012
%F A051945 G.f.: (2 - 3*x - 2*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - _Ilya Gutkovskiy_, Jun 13 2017
%F A051945 From _Peter Bala_, Aug 23 2025: (Start)
%F A051945 a(n) = binomial(2*n, n) + 4*binomial(2*n, n-1) = A000984(n) + 4*A001791(n).
%F A051945 a(n) ~ 4^n * 5/sqrt(Pi*n). (End)
%F A051945 E.g.f.: exp(2*x)*((1 + 5*x)*BesselI(0, 2*x) - BesselI(1, 2*x) - 5*x*BesselI(2, 2*x)). - _Stefano Spezia_, Aug 29 2025
%t A051945 Table[CatalanNumber[n](5n+1),{n,0,30}] (* _Harvey P. Dale_, Jul 27 2020 *)
%o A051945 (PARI) a(n) = (5*n+1)*binomial(2*n, n)/(n+1) \\ _Michel Marcus_, Jul 12 2013
%o A051945 (Magma) [Catalan(n)*(5*n+1):n in [0..27] ]; // _Marius A. Burtea_, Jan 05 2020
%o A051945 (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 A051945 Column k=5 of A330965.
%Y A051945 Cf. A016813, A000108, A051924.
%K A051945 easy,nonn,changed
%O A051945 0,2
%A A051945 _Barry E. Williams_, Dec 20 1999
%E A051945 Terms a(21) and beyond from _Andrew Howroyd_, Jan 04 2020