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 A259550 #26 May 18 2025 07:55:36
%S A259550 1,2,42,1001,25194,653752,17298645,463991880,12570420330,343176898988,
%T A259550 9425842448792,260170725132045,7210477496434485,200519284375732896,
%U A259550 5592628786362932776,156375886125188595376,4382048530314336892010,123033460966787345446836
%N A259550 a(n) = C(5*n-1,2*n)/3, n > 0, a(0) = 1.
%H A259550 D. Kruchinin and V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
%H A259550 V. V. Kruchinin and D. V. Kruchinin, <a href="http://www.naturalspublishing.com/files/published/yzmx634l9k644k.pdf">Composita and its properties</a>, J. Analysis and Number Theory 2 (2014), 1-8.
%H A259550 V. V. Kruchinin and D. V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kruchinin/kruch9.html">A Generating Function for the Diagonal T_{2n,n} in Triangles</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.
%H A259550 D. V. Kruchinin, <a href="http://dx.doi.org/10.1186/s13662-014-0347-9">On solving some functional equations</a>, Advances in Difference Equations, Vol. 1 (2015), 1687-1847.
%F A259550 G.f.: A(x) = 1 + (x*B(x)')/(B(x)), B(x) = (1 + x*B(x)^5)*C(x*B(x)^5) is g.f. of A060941, C(x) is g.f. of Catalan numbers.
%F A259550 a(n) = n*Sum_{i = 0..n}((C(5*n,i)*C(7*n-2*i-1,n-i))/(6*n-i)), n > 1, a(0) = 1.
%F A259550 a(n) = 1/5*A001450(n) for n >= 1. exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 23*x^2 + 377*x^3 + ... is the o.g.f. for the sequence of Duchon numbers A060941. - _Peter Bala_, Oct 05 2015
%F A259550 D-finite with recurrence 6*n*(3*n-1)*(2*n-1)*(3*n-2)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - _R. J. Mathar_, Nov 22 2024
%t A259550 Join[{1}, Table[Binomial[5 n - 1, 2 n]/3, {n, 30}]] (* _Vincenzo Librandi_, Jul 01 2015 *)
%o A259550 (Maxima)
%o A259550 makelist(if n=0 then 1 else binomial(5*n-1,2*n)/3,n,0,20);
%o A259550 (PARI) vector(20, n, n--; if (n==0, 1, binomial(5*n-1,2*n)/3)) \\ _Michel Marcus_, Jul 01 2015
%o A259550 (Magma) [1] cat [Binomial(5*n-1, 2*n)/3: n in [1..20]]; // _Vincenzo Librandi_, Jul 01 2015
%Y A259550 Cf. A000108, A167422, A001450, A060941.
%K A259550 nonn,easy
%O A259550 0,2
%A A259550 _Vladimir Kruchinin_, Jun 30 2015