A357508 a(n) = binomial(4*n,2*n) - 2*binomial(4*n,n).

-1, -2, 14, 484, 9230, 153748, 2434964, 37748520, 580043790, 8886848740, 136151207764, 2088760285456, 32108266614164, 494648505828904, 7637081136832840, 118158193386475984, 1831647087068431374, 28444051172077725444, 442429676097305612324

Offset: 0

Author

Peter Bala, Oct 01 2022

Keywords

• sign
• easy

Comments

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

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.]

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

Links

• C. Helou and G. Terjanian, On Wolstenholme’s theorem and its converse, J. Number Theory 128 (2008), 475-499.
• Z.-W. Sun and D. Wan, On Fleck quotients, arXiv:math/0603462 [math.NT], 2006-2007.

Crossrefs

Cf. A001448, A005810, A357509.

Programs

Maple

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

Formula

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

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