%I #48 Nov 12 2022 05:53:58
%S 0,1,3,13,67,376,2211,13378,82499,515659,3255628,20714354,132611491,
%T 853226921,5512508382,35739673513,232405291587,1515159860388,
%U 9900216370689,64816750480666,425097621975692,2792332673312356,18367642416256334,120972943783673953
%N Expansion of -x*d(log((1-x*(2/sqrt(3*x)) * sin((1/3) * arcsin(sqrt(27*x/4))))))/dx.
%H G. C. Greubel, <a href="/A242798/b242798.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry1/barry242.html">On the Central Coefficients of Riordan Matrices</a>, Journal of Integer Sequences, 16 (2013), #13.5.1.
%F a(n) = n*Sum_{k=1..n} binomial(3*n-2*k-1,n-k)/(2*n-k).
%F G.f.: x*(x*F'(x)+F(x))/(1-x*F(x)), where F(x) is g.f. of A001764.
%F D-finite with recurrence: 2*(n-1)*(2*n-1)*(91*n^3 - 531*n^2 + 962*n - 516)*a(n) = (2821*n^5 - 21921*n^4 + 62005*n^3 - 75435*n^2 + 33274*n - 24)*a(n-1) - (2821*n^5 - 21921*n^4 + 62005*n^3 - 75435*n^2 + 33274*n - 24)*a(n-2) + 3*(3*n - 8)*(3*n - 7)*(91*n^3 - 258*n^2 + 173*n + 6)*a(n-3). - _Vaclav Kotesovec_, Sep 21 2015
%F a(n) ~ 3^(3*n-1/2) / (7 * sqrt(Pi*n) * 4^n). - _Vaclav Kotesovec_, Sep 21 2015
%F From _Peter Luschny_, Jan 25 2019: (Start)
%F a(n) = (n/(2*n-1))*C(3*n-3, n-1)*(3F2)([1, 1-2*n, 1-n], [3/2-3*n/2, 2-3*n/2], 1/4).
%F a(n) = [x^n] (2/(1 + sqrt(1 - 4*x)))^n*(x/(1 - x)). (End)
%p ogf := n -> ((1 - sqrt(1 - 4*x))/(2*x))^n*x/(1 - x):
%p ser := n -> series(ogf(n), x, 46):
%p seq(coeff(ser(n), x, n), n=0..23); # _Peter Luschny_, Jan 25 2019
%t Table[n*Sum[Binomial[3*n - 2*k - 1, n - k]/(2*n - k), {k, 1, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 21 2015 *)
%t Table[(n Binomial[3 n - 3, n - 1] HypergeometricPFQ[{1, 1 - 2 n, 1 - n}, {3/2 - (3 n)/2, 2 - (3 n)/2}, 1/4])/(2 n - 1), {n, 0, 23}] (* _Peter Luschny_, Jan 25 2019 *)
%o (Maxima)
%o a(n):=n*sum(binomial(3*n-2*k-1,n-k)/(2*n-k),k,1,n);
%o (PARI) a(n) = n*sum(k=1, n, binomial(3*n-2*k-1,n-k)/(2*n-k)); \\ _Michel Marcus_, Nov 12 2022
%Y Cf. A001764, A174687.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, May 22 2014
%E Name edited by _Michel Marcus_, Jan 26 2019