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A363869
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a(n) = A108625(3*n, 2*n).
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6
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1, 55, 12559, 3685123, 1205189519, 418856591055, 151353475289275, 56193989426243199, 21283943385478109071, 8185785098679048061837, 3186604888590691870779559, 1252744279186835597251089055, 496508748101370063304243706939, 198134918989716743103591120933103
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OFFSET
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0,2
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COMMENTS
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a(n) = B(3*n, 2*n, 3*n) in the notation of Straub, equation 24. It follows from Straub, Theorem 3.2, that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
More generally, for positive integers r and s the sequence {A108625(r*n, s*n) : n >= 0} satisfies the same supercongruences.
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LINKS
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FORMULA
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a(n) = Sum_{k = 0..2*n} binomial(3*n, k)^2 * binomial(5*n-k, 3*n).
a(n) = Sum_{k = 0..2*n} (-1)^k * binomial(3*n, k)*binomial(5*n-k, 3*n)^2.
a(n) = hypergeom( [-2*n, -3*n, 3*n+1], [1, 1], 1).
a(n) = [x^(2*n)] 1/(1 - x)*Legendre_P(3*n, (1 + x)/(1 - x)).
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MAPLE
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A108625 := (n, k) -> hypergeom([-n, -k, n+1], [1, 1], 1):
seq(simplify(A108625(3*n, 2*n)), n = 0..20);
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MATHEMATICA
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A363869[n_] := HypergeometricPFQ[{-2*n, -3*n, 3*n + 1}, {1, 1}, 1];
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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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