A357508 - OEIS

%I #23 Mar 18 2023 08:49:14

%S -1,-2,14,484,9230,153748,2434964,37748520,580043790,8886848740,

%T 136151207764,2088760285456,32108266614164,494648505828904,

%U 7637081136832840,118158193386475984,1831647087068431374,28444051172077725444,442429676097305612324

%N a(n) = binomial(4*n,2*n) - 2*binomial(4*n,n).

%C Sun and Wan's supercongruence stated below apparently generalizes as follows:

%C Let m be an integer and k a positive integer. Define u(n) = binomial((m+2)*n,(k+1)*n) - binomial(m,k)*binomial((m+2)*n,n). We conjecture that u(n) == u(1) (mod p^5) for all primes p >= 7. [added 22 Oct 2022: the conjecture is true: apply Helou and Terjanian, Section 3, Proposition 2.]

%C Conjecture: for r >= 2, u(p^r) == u(p^(r-1)) ( mod p^(3*r+3) ) for all primes p >= 5. - _Peter Bala_, Oct 13 2022

%H 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 Z.-W. Sun and D. Wan, <a href="https://arxiv.org/abs/math/0603462">On Fleck quotients</a>, arXiv:math/0603462 [math.NT], 2006-2007.

%F a(n) = A001448(n) - 2*A005810(n).

%F a(p) == -2 (mod p^5) for all primes p >= 7. (Sun and Wan, Corollary 1.5.)

%p seq(binomial(4*n,2*n) - 2*binomial(4*n,n), n = 0..20);

%Y Cf. A001448, A005810, A357509.

%K sign,easy

%O 0,2

%A _Peter Bala_, Oct 01 2022