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A188675 Partial sums of the binomial coefficients binomial(3*n,n) (A005809).

%I A188675 #26 Sep 14 2024 19:01:44
%S A188675 1,4,19,103,598,3601,22165,138445,873916,5560741,35605756,229142476,
%T A188675 1480820176,9603245620,62463474700,407330900284,2662179813931,
%U A188675 17433248900656,114359597479261,751343566800961,4943188072606456
%N A188675 Partial sums of the binomial coefficients binomial(3*n,n) (A005809).
%H A188675 G. C. Greubel, <a href="/A188675/b188675.txt">Table of n, a(n) for n = 0..1000</a>(terms 1..100 from Vincenzo Librandi)
%F A188675 a(n) = Sum_{k=0..n} binomial(3*k,k).
%F A188675 Recurrence: 2*(n+2)*(2n+3)*a(n+2)-(31*n^2+95*n+72)*a(n+1)+3*(3*n+4)(3*n+5)*a(n)=0.
%F A188675 G.f.: 2*cos((1/3)*arcsin(3*sqrt(3*x)/2))/((1-x)*sqrt(4-27*x)).
%F A188675 a(n) ~ sqrt(3)*27^(n+1)/(46*4^n*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 20 2012
%t A188675 Table[Sum[Binomial[3k, k], {k, 0, n}], {n, 0, 20}]
%t A188675 Accumulate[Table[Binomial[3n,n],{n,0,20}]] (* Nearly 300 times faster than the program above. *) (* _Harvey P. Dale_, Sep 14 2024 *)
%o A188675 (Maxima) makelist(sum(binomial(3*k,k),k,0,n),n,0,20);
%o A188675 (PARI) for(n=0,25, print1(sum(k=0,n, binomial(3*k,k)), ", ")) \\ _G. C. Greubel_, Jan 27 2017
%Y A188675 Cf. A001764, A005809, A104859, A188676, A188678 - A188687.
%Y A188675 Cf. A263134: Sum_{k=0..n} binomial(3*k+1,k).
%Y A188675 Cf. A087413: Sum_{k=0..n} binomial(3*k+2,k).
%K A188675 nonn,easy
%O A188675 0,2
%A A188675 _Emanuele Munarini_, Apr 08 2011