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 A051944 #34 Aug 23 2025 09:19:51
%S A051944 1,5,18,65,238,882,3300,12441,47190,179894,688636,2645370,10192588,
%T A051944 39373700,152443080,591385545,2298248550,8945490510,34867625100,
%U A051944 136079265630,531693754020,2079632696700,8141948163960,31904544069450,125120702290428,491056586546652
%N A051944 a(n) = C(n)*(4*n+1) where C(n) = Catalan numbers (A000108).
%D A051944 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A051944 Andrew Howroyd, <a href="/A051944/b051944.txt">Table of n, a(n) for n = 0..200</a>
%F A051944 The Hankel determinant transform is A025172(n-1). - _Michael Somos_, Sep 17 2006
%F A051944 -(n+1)*(4*n-3)*a(n) + 2*(4*n+1)*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Nov 19 2014
%F A051944 G.f.: (3 - 4*x - 3*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - _Ilya Gutkovskiy_, Jun 13 2017
%F A051944 From _Peter Bala_, Aug 23 2025: (Start)
%F A051944 a(n) = binomial(2*n, n) + 3*binomial(2*n, n-1) = A000984(n) + 3*A001791(n).
%F A051944 a(n) ~ 4^(n+1)/sqrt(Pi*n). (End)
%t A051944 Table[CatalanNumber[n](4n+1),{n,0,30}] (* _Harvey P. Dale_, Feb 21 2022 *)
%o A051944 (PARI) {a(n)=if(n<0, 0, (4*n+1)*binomial(2*n,n)/(n+1))} /* _Michael Somos_, Sep 17 2006 */
%o A051944 (Magma) [Catalan(n)*(4*n+1):n in [0..30] ]; // _Marius A. Burtea_, Jan 05 2020
%o A051944 (Magma) R<x>:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (3 - 4*x - 3*Sqrt(1 - 4*x))/(2*x*Sqrt(1 - 4*x)))) ); // _Marius A. Burtea_, Jan 05 2020
%Y A051944 Column k=4 of A330965.
%Y A051944 Cf. A016777, A000108, A051924.
%K A051944 easy,nonn,changed
%O A051944 0,2
%A A051944 _Barry E. Williams_, Dec 20 1999
%E A051944 Terms a(21) and beyond from _Andrew Howroyd_, Jan 02 2020