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A357569 a(n) = binomial(3*n,n)^2 - 27*binomial(2*n,n).

%I A357569 #17 Jul 07 2024 21:08:01
%S A357569 -26,-45,63,6516,243135,9011205,344597148,13520945736,540917244351,
%T A357569 21966327267885,902702921361813,37456461969311736,1566697064604277788,
%U A357569 65973795093057780936,2794203818388994498200,118933541228931589568016,5084343623375039833670079,218184481964802862563857685
%N A357569 a(n) = binomial(3*n,n)^2 - 27*binomial(2*n,n).
%C A357569 Conjectures:
%C A357569 1) a(p^r) == a(p^(r-1)) ( mod p^(3*r+3) ) for r >= 2 and all primes p >= 3.
%C A357569 These are stronger supercongruences than those satisfied separately by the sequences {binomial(2*n,n)} = A000984 and {binomial(3*n,n)} = A005809.
%C A357569 2) More generally, for k >= 1, the sequence {2*binomial(3*n,n)^k - k*(3^(k+1))*binomial(2*n,n): n >= 0} may satisfy the same supercongruences. This is the case k = 2. See A357509 for the case k = 1.
%H A357569 C. Helou and G. Terjanian, <a href="https://doi.org/10.1016/j.jnt.2007.06.008">On Wolstenholme’s theorem and its converse</a>, J. Number Theory 128 (2008), 475-499.
%H A357569 Romeo Meštrović, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv preprint arXiv:1111.3057 [math.NT], 2011.
%F A357569 a(n) = 3*A188662(n) - 27*A000984(n) = 3*A005809(n)^2 - 27*A000984(n).
%F A357569 a(k*p^r) == a(k*p^(r-1)) ( mod p^(3*r) ) for positive integers k and r and for all primes p >= 5 (see Meštrović, Section 6, equation 39).
%F A357569 a(p) == a(1) (mod p^5) for all primes p >= 7 (apply Helou and Terjanian, Section 3, Proposition 2).
%e A357569 Examples of supercongruences:
%e A357569 a(13) - a(1) = 65973795093057780936 + 45 = (3^2)*(13^5)*163*121122434651 == 0 (mod 13^5).
%e A357569 a(5^2) - a(5) = 2765555290416839473031163791322085183080 - 9011205 = (3^2)*(5^9)* 229*2333*6840413*74974087*574203805501 == 0 (mod 5^9).
%p A357569 seq(binomial(3*n,n)^2 - 27*binomial(2*n,n), n = 0..20);
%t A357569 Table[Binomial[3n,n]^2-27*Binomial[2n,n],{n,0,30}] (* _Harvey P. Dale_, Jun 12 2023 *)
%Y A357569 Cf. A000984, A005809, A188662, A357509, A357567, A357568, A357955.
%K A357569 sign,easy
%O A357569 0,1
%A A357569 _Peter Bala_, Oct 21 2022