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 A066381 #57 Aug 17 2025 09:53:31
%S A066381 1,5,37,299,2517,21700,190051,1683218,15033173,135142796,1221246132,
%T A066381 11083374659,100946732307,922205369324,8446802334994,77542088287444,
%U A066381 713250450657109,6572130378649468,60652194138406780,560522209086365852
%N A066381 a(n) = Sum_{k=0..n} binomial(4*n,k).
%H A066381 Harry J. Smith, <a href="/A066381/b066381.txt">Table of n, a(n) for n = 0..150</a>
%F A066381 G.f.: s/((s-2)*(3*s-4)) where s = o.g.f.(A002293) which satisfies 1-s+x*s^4 = 0. - _Mark van Hoeij_, May 05 2013
%F A066381 a(0) = 1, a(n) = 16*a(n-1)-2*(44*n^3-34*n^2-2*n+3)*(4*n-4)!/(n!*(3*n-1)!). - _Tani Akinari_, Sep 02 2014
%F A066381 a(n) are special values of the hypergeometric function 2F1, in Maple notation: a(n)=16^n-binomial(4*n,n+1)*hypergeom([1,-3*n+1],[n+2],-1), n=0,1,... . - _Karol A. Penson_, Jun 03 2015
%F A066381 a(n) ~ (256/27)^n * sqrt(3/(2*Pi*n)). - _Vaclav Kotesovec_, Jun 03 2015
%F A066381 a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(3*n)). - _Ilya Gutkovskiy_, Oct 25 2017
%F A066381 a(n) = Sum_{k=0..floor(n/2)} binomial(4*n+1,n-2*k). - _Seiichi Manyama_, Apr 09 2024
%F A066381 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+k-1,k). - _Seiichi Manyama_, Jul 30 2025
%F A066381 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - _Seiichi Manyama_, Aug 08 2025
%F A066381 G.f.: 1/(1 - x*g^3*(8-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 17 2025
%p A066381 ogf := eval(s/((s-2)*(3*s-4)), s = RootOf(1-s+x*s^4, s));
%p A066381 series(ogf, x=0, 25); # _Mark van Hoeij_, May 05 2013
%t A066381 Table[Sum[Binomial[4*n,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 03 2015 *)
%o A066381 (PARI) { for (n=0, 150, a=0; for (k=0, n, a+=binomial(4*n, k)); write("b066381.txt", n, " ", a) ) } \\ _Harry J. Smith_, Feb 12 2010
%o A066381 (Maxima) a[0]:1$ a[1]:5$ a[n]:=8*((3784*n^6-18764*n^5+34432*n^4 -28138*n^3+9028*n^2-24*n-315)*a[n-1]+16*(3-2*n)*(4*n-5)*(4*n-7)*(44*n^3-34*n^2-2*n+3)*a[n-2])/(3*n*(3*n-1)*(3*n-2)*(44*n^3-166*n^2 +198*n-73))$ makelist(a[n],n,0,1000); /* _Tani Akinari_, Sep 02 2014 */
%Y A066381 Cf. A002293, A032443, A066380, A371739.
%K A066381 nonn
%O A066381 0,2
%A A066381 _N. J. A. Sloane_, Dec 23 2001