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%I #18 Mar 22 2022 05:53:59
%S 5,56,420,2640,15015,80080,408408,2015520,9699690,45762640,212469400,
%T 973496160,4411154475,19800295200,88158457200,389753179200,
%U 1712478031110,7483097278800,32540135136600,140883148005600,607558575774150,2610765994183776,11182476723339600
%N Third derivative of Catalan generating function/3!.
%H Andrew Howroyd, <a href="/A030060/b030060.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = (2*(n+3))!/(3!*n!*(n+4)!) = (n+1)*(n+2)*(n+3)*C(n+3)/6, C(n): Catalan numbers.
%F From _Amiram Eldar_, Mar 22 2022: (Start)
%F Sum_{n>=0} 1/a(n) = (sqrt(3)*Pi - 5)/2.
%F Sum_{n>=0} (-1)^n/a(n) = 9*sqrt(5)*log(phi) - 19/2, where phi is the golden ratio (A001622). (End)
%t Array[CatalanNumber[#] Binomial[#, 3] &, 19, 3] (* _Michael De Vlieger_, Dec 17 2017 *)
%o (MuPAD) combinat::catalan(n) *binomial(n,3) $ n = 3..21 // _Zerinvary Lajos_, Feb 15 2007
%o (PARI) a(n) = (2*(n+3))!/(3!*n!*(n+4)!) \\ _Andrew Howroyd_, Dec 17 2017
%Y Cf. A000108, A001622.
%K nonn
%O 0,1
%A _Wolfdieter Lang_