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A357508
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a(n) = binomial(4*n,2*n) - 2*binomial(4*n,n).
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4
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-1, -2, 14, 484, 9230, 153748, 2434964, 37748520, 580043790, 8886848740, 136151207764, 2088760285456, 32108266614164, 494648505828904, 7637081136832840, 118158193386475984, 1831647087068431374, 28444051172077725444, 442429676097305612324
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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a(p) == -2 (mod p^5) for all primes p >= 7. (Sun and Wan, Corollary 1.5.)
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MAPLE
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seq(binomial(4*n, 2*n) - 2*binomial(4*n, n), n = 0..20);
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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