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%I #32 Oct 14 2015 06:56:56
%S 1,6,34,199,1200,7388,46148,291305,1853580,11868585,76380825,
%T 493606725,3201081873,20821158233,135776966761,887393271310,
%U 5811082966885,38119865826420,250447855600320,1647729357535485,10854207824989830,71581930485576630
%N a(n) = Sum_{k=1..n} C(3*k,k)/3.
%H Vincenzo Librandi, <a href="/A087413/b087413.txt">Table of n, a(n) for n = 1..200</a>
%F G.f.: 1/((3*g-1)*(g^3-2*g^2+g-1)*(g-1)^2) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 10 2011
%F Recurrence: 2*n*(2*n-1)*a(n) = (31*n^2-29*n+6)*a(n-1) - 3*(3*n-2)*(3*n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 14 2012
%F a(n) ~ 3^(3*n+5/2)/(23*2^(2*n+1)*sqrt(Pi)*sqrt(n)). - _Vaclav Kotesovec_, Oct 14 2012
%F a(n) = Sum_{k=1..n} binomial(3*k-1,k-1). [_Bruno Berselli_, Oct 10 2015]
%t Table[Sum[Binomial[3*k,k]/3,{k,1,n}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%o (PARI) a(n)=sum(k=1,n,binomial(3*k,k))/3 \\ _Charles R Greathouse IV_, Nov 10 2011
%o (PARI) a=vector(99,i,1);for(n=2,#a,a[n]=a[n-1]+binomial(3*n,n)/3);a \\ _Charles R Greathouse IV_, Nov 10 2011
%Y Cf. A188675: Sum_{k=0..n} binomial(3*k,k).
%Y Cf. A263134: Sum_{k=0..n} binomial(3*k+1,k).
%K nonn,easy
%O 1,2
%A _Benoit Cloitre_, Oct 21 2003
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