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a(n) = Sum_{k=1..prime(n)^2-1} binomial(2k,k).
2

%I #8 Jul 09 2018 23:19:59

%S 28,17576,43308802158650,8610524734277600186228691452,

%T 121374542758943982922417964798154019940274699584207321286055873543631298,

%U 8126392396649531937838689708830356413772063825711016912849229977138431439363305375418692100492504264

%N a(n) = Sum_{k=1..prime(n)^2-1} binomial(2k,k).

%H D. Callan, <a href="https://www.jstor.org/stable/40391137">Divisibility of a Central Binomial Sum: Problem A11292 and A11307</a>, Amer. Math. Monthly, 116 (2009), 468-470.

%t Table[Sum[Binomial[2k,k],{k,Prime[n]^2-1}],{n,7}] (* _Harvey P. Dale_, Dec 26 2014 *)

%o (PARI) a(n) = sum(k=1, prime(n)^2-1, binomial(2*k,k)); \\ _Michel Marcus_, Jul 07 2018

%Y Cf. A146977, A066796, A147291.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Apr 25 2009