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%I #13 Jun 05 2017 19:06:16
%S 1,3,12,54,265,1401,7903,47088,293319,1892440,12548041,84988566,
%T 585314652,4085026386,28820064810,205156454376,1471492171068,
%U 10622954509803,77122189800121,562684397212060,4123449352097229,30336562360256955
%N Sum_{k=0..n} ((binomial(3*k,k)*binomial(2*n-k,n))/(2*k+1)).
%H G. C. Greubel, <a href="/A270489/b270489.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: P(C(x))/(1-x/(1-C(x)))^2/x, where C(x)=(1-sqrt(1-4*x))/2, P(x)/x is g.f. of A001764.
%F Recurrence: 46*(n-2)*(n-1)*n*(2*n + 1)*(133*n - 239)*a(n) = 5*(n-2)*(n-1)*(38969*n^3 - 108996*n^2 + 78733*n - 18774)*a(n-1) - 4*(n-2)*(242858*n^4 - 1286417*n^3 + 2496793*n^2 - 2103937*n + 643755)*a(n-2) + 36*(3*n - 7)*(3*n - 5)*(6*n - 13)*(6*n - 11)*(133*n - 106)*a(n-3). - _Vaclav Kotesovec_, Mar 18 2016
%F a(n) ~ 3^(6*n + 7/2) / (19^(3/2) * sqrt(Pi) * 2^(2*n+2) * 23^(n - 1/2) * n^(3/2)). - _Vaclav Kotesovec_, Mar 18 2016
%t Table[Sum[Binomial[3*k,k]*Binomial[2*n-k,n]/(2*k+1), {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 18 2016 *)
%o (Maxima)
%o taylor((sqrt(2)*sin(asin((3^(3/2)*sqrt(1-sqrt(1-4*x)))/2^(3/2))/3)*
%o sqrt(1-sqrt(1-4*x)))/(sqrt(3)*(1-x/(1-(1-sqrt(1-4*x))/2)^2))/x,x,0,20);
%o (PARI) for(n=0,25, print1(sum(k=0,n, binomial(3*k,k)*binomial(2*n-k,n)/(2*k+1)), ", ")) \\ _G. C. Greubel_, Jun 05 2017
%Y Cf. A000108, A001764, A092392.
%K nonn
%O 0,2
%A _Vladimir Kruchinin_, Mar 18 2016
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