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%I #26 Jul 26 2021 01:49:02
%S 1,0,3,9,46,227,1201,6551,36712,209963,1220752,7193888,42873220,
%T 257957352,1564809168,9559946496,58768808463,363261736872,
%U 2256369305793,14076552984507,88163556913188,554148894304557,3494365949734563
%N Alternate partial sums of binomial(3*n,n)/(2*n+1).
%H Seiichi Manyama, <a href="/A188678/b188678.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from Vincenzo Librandi)
%F a(n) = Sum_{k=0..n} binomial(3*k,k)*(-1)^(n-k)/(2*k+1).
%F Recurrence: 2*(2*n^2+9*n+10)*a(n+2)-(23*n^2+63*n+40)*a(n+1)-3*(9*n^2+27*n+20)*a(n)=0.
%F G.f.: 2*sin((1/3)*arcsin(3*sqrt(3*x)/2))/((1+x)*sqrt(3*x)).
%F a(n) ~ 3^(3*n+3+1/2)/(31*sqrt(Pi)*n^(3/2)*2^(2*n+2)). - _Vaclav Kotesovec_, Aug 06 2013
%F G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^2 * A(x)^3. - _Ilya Gutkovskiy_, Jul 25 2021
%t Table[Sum[Binomial[3k,k](-1)^(n-k)/(2k+1),{k,0,n}],{n,0,20}]
%o (Maxima) makelist(sum(binomial(3*k,k)*(-1)^(n-k)/(2*k+1),k,0,n),n,0,20);
%Y Cf. A005809, A001764, A188676, A104859, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.
%K nonn,easy
%O 0,3
%A _Emanuele Munarini_, Apr 08 2011
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