%I #20 Sep 08 2022 08:46:16
%S 0,27,2250,105987,191420427,8091223647,14764068268068,636877933530303,
%T 1202912275541006163,101416777765228668135243,
%U 4470228055779262703971971,387215990724249198145972139532,763815804413169191825705670410076,33986135421741862065520464883016403
%N a(n) = binomial(3*prime(n), prime(n)) - 3*binomial(2*prime(n), prime(n)) + 3.
%H Vincenzo Librandi, <a href="/A272630/b272630.txt">Table of n, a(n) for n = 1..200</a>
%H Chun-Gang Ji, <a href="http://www.ams.org/journals/proc/2005-133-12/S0002-9939-05-07939-6/">A simple proof of a curious congruence by Zhao</a>, Proc. Amer. Math. Soc. 133 (2005).
%F Sum of binomial(prime(n), i)*binomial(prime(n), j)*binomial(prime(n), k) where i+j+k = prime(n) and i,j,k > 0 (see combinatorial identity on page 3471 in Chun-Gang Ji's paper).
%t Table[Binomial[3 Prime[n], Prime[n]] - 3 Binomial[2 Prime[n], Prime[n]] + 3, {n, 20}]
%t Binomial[3#,#]-3Binomial[2#,#]+3&/@Prime[Range[20]] (* _Harvey P. Dale_, Jul 29 2021 *)
%o (Magma) [Binomial(3*p,p)-3*Binomial(2*p,p)+3: p in PrimesUpTo(50)];
%o (PARI) lista(nn) = {forprime(p=2, nn, print1(binomial(3*p,p) - 3*binomial(2*p,p) + 3, ", "));} \\ _Altug Alkan_, May 05 2016
%K nonn
%O 1,2
%A _Vincenzo Librandi_, May 05 2016