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A357508
a(n) = binomial(4*n,2*n) - 2*binomial(4*n,n).
4
-1, -2, 14, 484, 9230, 153748, 2434964, 37748520, 580043790, 8886848740, 136151207764, 2088760285456, 32108266614164, 494648505828904, 7637081136832840, 118158193386475984, 1831647087068431374, 28444051172077725444, 442429676097305612324
OFFSET
0,2
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.
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.)
MAPLE
seq(binomial(4*n, 2*n) - 2*binomial(4*n, n), n = 0..20);
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Peter Bala, Oct 01 2022
STATUS
approved