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A364516
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a(n) = (2/3) * (8*n)!*(3*n)!^2/((6*n)!*(4*n)!*(2*n)!*n!^2) for n >= 1, with a(0) = 1.
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2
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1, 28, 3900, 685216, 133501500, 27583083528, 5919115212192, 1304298034300800, 293086491979934268, 66857471357130883000, 15434267149448839091400, 3597756971630997935635200, 845406187463509505329860000, 200002748013094535687584437696
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = [x^n] (1 - x)^(2*n) * Legendre_P(6*n-1, (1 + x)/(1 - x)) for n >= 1.
a(n) = Sum_{k = 0..n} binomial(6*n - 1, n - k)^2 * binomial(4*n + k - 2, k).
a(n) = (6*n-1)!*(4*n-1/2)!*(2*n-1/2)!/((4*n-1)! * (3*n-1/2)!^2 * n!^2) for n >= 1 (fractional factorials are defined in terms of the gamma function, for example, (4*n - 1/2)! = gamma(4*n + 1/2)).
a(n) ~ 2^(8*n) * sqrt(6)/(6*Pi*n).
P-recursive: a(0) = 1; for n >= 1, a(n) = (8*n-1)*(8*n-3)*(8*n-5)*(8*n-7)*(3*n-1)*(3*n-2)/((6*n-1)*(6*n-5)*(2*n-1)^2*n^2) * a(n-1) with a(1) = 28.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.
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EXAMPLE
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Examples of supercongruences:
a(7) - a(1) = 1304298034300800 - 28 = (2^2)*(7^4)*103553*1311481 == 0 (mod 7^4).
a(11) - a(1) = 3597756971630997935635200 - 28 = (2^2)*(11^3)*22567*7702811* 3887502719 == 0 (mod 11^3).
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MAPLE
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seq( (2/3) * (8*n)!*(3*n)!^2/((6*n)!*(4*n)!*(2*n)!*n!^2), n = 0..15);
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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