Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #45 Apr 09 2024 11:45:40
%S 1,5,37,299,2517,21700,190051,1683218,15033173,135142796,1221246132,
%T 11083374659,100946732307,922205369324,8446802334994,77542088287444,
%U 713250450657109,6572130378649468,60652194138406780,560522209086365852
%N a(n) = Sum_{k=0..n} binomial(4*n,k).
%H Harry J. Smith, <a href="/A066381/b066381.txt">Table of n, a(n) for n = 0..150</a>
%F 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 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 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 a(n) ~ (256/27)^n * sqrt(3/(2*Pi*n)). - _Vaclav Kotesovec_, Jun 03 2015
%F a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(3*n)). - _Ilya Gutkovskiy_, Oct 25 2017
%F a(n) = Sum_{k=0..floor(n/2)} binomial(4*n+1,n-2*k). - _Seiichi Manyama_, Apr 09 2024
%p ogf := eval(s/((s-2)*(3*s-4)), s = RootOf(1-s+x*s^4, s));
%p series(ogf, x=0, 25); # _Mark van Hoeij_, May 05 2013
%t Table[Sum[Binomial[4*n,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 03 2015 *)
%o (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 (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 Cf. A002293, A032443, A066380, A371739.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Dec 23 2001