cp's OEIS Frontend

A375180 a(n) = Sum_{k = 0..n-1} (-1)^(n+k+1)*binomial(3*n, k)^3.

%I A375180 #10 Aug 14 2024 08:36:59
%S A375180 0,1,215,45928,10362231,2450260001,600869373182,151570671244560,
%T A375180 39096342054496887,10267275084850974619,2736324289110748127715,
%U A375180 738255282011665067114400,201254884472471159485086750,55352399437924814524429123488,15341068552569688728602977821596
%N A375180 a(n) = Sum_{k = 0..n-1} (-1)^(n+k+1)*binomial(3*n, k)^3.
%C A375180 Compare with the identity Sum_{k = 0..n-1} (-1)^(n+k+1)*binomial(3*n, k) = (1/3) * binomial(3*n, n) = (1/3) * A005809(n) for n >= 1.
%C A375180 The binomial coefficients satisfy the supercongruence (1/3) * binomial(3*p, p) == 1 (mod p^3) for all primes p >= 5 (Meštrović, Equation 35).
%C A375180 We conjecture that for the present sequence the stronger supercongruence a(p) == 1 (mod p^5) holds for all primes p >= 7.
%C A375180 Further, we conjecture that for r >= 2 and prime p >= 5, a(p^r) == a(p^(r-1)) (mod p^(3*r+3)).
%C A375180 More generally, for a positive integer m, define a sequence {b_m(n) : n >= 0} by setting b_m(n) = Sum_{k = 0..n-1} (-1)^(n+k+1)*binomial(3*n, k)^(2*m+1). We conjecture that the supercongruence b_m(p) == 1 (mod p^(2*m+3)) holds for all primes p >= 2*m + 5, and for r >= 2, the supercongruence b_m(p^r) == b_m(p^(r-1)) (mod p^(3*r+2*m+1)) also holds for all primes p >= 2*m + 5.
%H A375180 Romeo Meštrović, <a href="https://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012)</a>, arXiv:1111.3057 [math.NT], (2011).
%F A375180 a(n) ~ 3^(9*n - 1/2) / (Pi^(3/2) * n^(3/2) * 2^(6*n+3)). - _Vaclav Kotesovec_, Aug 08 2024
%e A375180 Examples of supercongruences:
%e A375180 a(7) - a(1) = 151570671244560 - 1 = (7^5)*379*2269*10487 == 0 (mod 7^5);
%e A375180 a(13) - a(1) = 55352399437924814524429123488 - 1 = (13^5)*149080105032749915900459 == 0 (mod 13^5).
%p A375180 seq(add( (-1)^(n+k+1)*binomial(3*n, k)^3, k = 0..n-1), n = 0..20);
%Y A375180 Cf. A005809, A375178, A375179.
%K A375180 nonn,easy
%O A375180 0,3
%A A375180 _Peter Bala_, Aug 05 2024