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A290059 a(n) = binomial(2*prime(n)-1, prime(n)-1) where prime(n) is the n-th prime.

%I A290059 #35 Mar 29 2024 14:32:05
%S A290059 3,10,126,1716,352716,5200300,1166803110,17672631900,4116715363800,
%T A290059 15033633249770520,232714176627630544,873065282167813104916,
%U A290059 212392290424395860814420,3318776542511877736535400,812850570172585125274307760,3136262529306125724764953838760
%N A290059 a(n) = binomial(2*prime(n)-1, prime(n)-1) where prime(n) is the n-th prime.
%C A290059 Charles Babbage (1791-1871) proved in 1819 that for every prime p > 2 this sequence is congruent to 1 (mod p^2).
%C A290059 Joseph Wolstenholme (1829-1891) proved in 1862 that for every prime p > 3 this sequence is congruent to 1 (mod p^3).
%H A290059 Seiichi Manyama, <a href="/A290059/b290059.txt">Table of n, a(n) for n = 1..261</a>
%H A290059 Charles Babbage, <a href="https://books.google.de/books?id=KrA-AAAAYAAJ&amp;pg=PA46">Demonstration of a theorem relating to prime numbers</a>, The Edinburgh philosophical journal 1 (1819), p. 46-49.
%H A290059 Joseph Wolstenholme, <a href="https://books.google.de/books?id=vL0KAAAAIAAJ&amp;pg=PA35">On certain properties of prime numbers</a>, The quarterly journal of pure and applied mathematics 5 (1862), p. 35-39.
%F A290059 log(a(n)) ~ 2*log(2)*n * (log(n) + log(log(n)) - 1). - _Vaclav Kotesovec_, May 07 2022
%p A290059 seq(binomial(2*ithprime(i)-1,ithprime(i)-1),i=1..16);
%t A290059 Array[Function[p, Binomial[2*p - 1, p - 1]]@ Prime@ # &, 16] (* _Michael De Vlieger_, Jul 19 2017 *)
%t A290059 Binomial[2#-1,#-1]&/@Prime[Range[20]] (* _Harvey P. Dale_, Mar 29 2024 *)
%o A290059 (PARI) a(n) = my(p=prime(n)); binomial(2*p-1, p-1); \\ _Michel Marcus_, Jul 19 2017
%o A290059 (Python)
%o A290059 from sympy import prime, binomial
%o A290059 def a(n):
%o A290059     p=prime(n)
%o A290059     return binomial(2*p - 1, p - 1)
%o A290059 print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Jul 20 2017
%Y A290059 Cf. A088164. Subsequence of A001700.
%K A290059 nonn
%O A290059 1,1
%A A290059 _Martin Renner_, Jul 19 2017