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 A066380 #65 Aug 17 2025 09:53:48
%S A066380 1,4,22,130,794,4944,31180,198440,1271626,8192524,53009102,344212906,
%T A066380 2241812648,14637774688,95786202688,628002401520,4124304597834,
%U A066380 27126202533252,178651732923346,1178005033926998,7776048412324714
%N A066380 a(n) = Sum_{k=0..n} binomial(3*n,k).
%D A066380 R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 425.
%H A066380 Stefano Spezia, <a href="/A066380/b066380.txt">Table of n, a(n) for n = 0..1200</a> (first 151 terms from Harry J. Smith)
%F A066380 a(n) ~ C(3n, n)(2 - 4/n + O(1/n^2)).
%F A066380 G.f.: (1-g)/((3*g-1)*(2*g-1)) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 10 2011
%F A066380 G.f.: x*(d/dx)log((F(x)-1)/(2-F(x))), where F(x) is g.f. of A001764. - _Vladimir Kruchinin_, Jun 13 2014
%F A066380 a(0)=1, a(n) = 8*a(n-1) - (5*n^2+n-2)*(3*n-3)!/((2*n-1)!*n!). - _Tani Akinari_, Sep 02 2014
%F A066380 a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(2*n)). - _Ilya Gutkovskiy_, Oct 25 2017
%F A066380 a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+1,n-2*k). - _Seiichi Manyama_, Apr 09 2024
%F A066380 a(n) = binomial(1+3*n, n)*hypergeom([1, (1-n)/2, -n/2], [1+n, 3/2+n], 1). - _Stefano Spezia_, Apr 09 2024
%F A066380 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+k-1,k). - _Seiichi Manyama_, Jul 30 2025
%F A066380 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - _Seiichi Manyama_, Aug 08 2025
%F A066380 From _Seiichi Manyama_, Aug 17 2025: (Start)
%F A066380 G.f.: 1/(1 - x*g^2*(6-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
%F A066380 G.f.: g/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. (End)
%p A066380 A066380:=n->add(binomial(3*n,k), k=0..n): seq(A066380(n), n=0..20); # _Wesley Ivan Hurt_, Sep 18 2014
%t A066380 Table[Sum[Binomial[3 n, k], {k, 0, n}], {n, 0, 20}] (* _Geoffrey Critzer_, May 27 2013 *)
%t A066380 a[n_] := 8^n - (2*n)/(n+1)*Binomial[3*n, n]*Hypergeometric2F1[1, -2*n+1, n+2, -1]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 02 2013 *)
%o A066380 (PARI) { for (n=0, 150, a=0; for (k=0, n, a+=binomial(3*n, k)); write("b066380.txt", n, " ", a) ) } \\ _Harry J. Smith_, Feb 12 2010
%o A066380 (Maxima) a[0]:1$ a[n]:=8*a[n-1]-(5*n^2+n-2)*(3*n-3)!/((2*n-1)!*n!)$ makelist(a[n],n,0,200); /* _Tani Akinari_, Sep 02 2014 */
%Y A066380 Cf. A001764, A032443, A066381, A371739.
%K A066380 nonn,easy
%O A066380 0,2
%A A066380 _N. J. A. Sloane_, Dec 23 2001