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 A188676 Alternate partial sums of the binomial coefficients binomial(3*n,n). 11

%I

%S 1,2,13,71,424,2579,15985,100295,635176,4051649,25993366,167543354,

%T 1084134346,7038291098,45821937982,299045487602,1955803426045,

%U 12815265660680,84111082917925,552872886403775,3638971619401720

%N Alternate partial sums of the binomial coefficients binomial(3*n,n).

%H Vincenzo Librandi, <a href="/A188676/b188676.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = sum(k=0..n, (-1)^(n-k)*binomial(3*k,k) ).

%F Recurrence: 2*(n+2)*(2n+3)*a(n+2)-(23*n^2+67*n+48)*a(n+1)-3*(3*n+4)*(3n+5)*a(n)=0.

%F G.f.: 2*cos((1/3)*arcsin(3*sqrt(3*x)/2))/((1+x)*sqrt(4-27*x)).

%F a(n) ~ 3^(3*n+7/2)/(62*4^n*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 20 2012

%t Table[Sum[Binomial[3k, k](-1)^(n-k), {k, 0, n}], {n, 0, 20}]

%o (Maxima) makelist(sum(binomial(3*k,k)*(-1)^(n-k),k,0,n),n,0,20);

%Y Cf. A005809, A001764, A104859, A188678, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.

%K nonn,easy

%O 0,2

%A _Emanuele Munarini_, Apr 08 2011

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Last modified September 23 16:17 EDT 2020. Contains 337314 sequences. (Running on oeis4.)